def CalibrationHa(z,oldLum,it): newLum = [0. for x in range((it.shape[0]))] oldpara = np.array([[0. for i in range(3)] for i in range(it.shape[0])]) oldLF = np.array([0. for i in range(it.shape[0])]) interpolatingSize =100 for i in range(oldpara.shape[0]): oldpara[i][0] = 1.37e-3 oldpara[i][2] = -1.35 if z[it[i]] <1.3 :oldpara[i][1] = 5.1e41*np.power((1+z[it[i]]),3.1) if z[it[i]] >=1.3:oldpara[i][1] =6.8e42 oldLF[i] = LuminosityFunction(oldpara[i],oldLum[i]) rangea = oldLF.copy() rangea.sort() rangea = rangea[rangea>0] grid = np.array([[0. for i in range(interpolatingSize)] for i in range(2)]) mpgrid = mpmath.arange(mpmath.log10(rangea.min()),mpmath.log10(rangea.max()),(np.log10(rangea.max())-np.log10(rangea.min()))/interpolatingSize) grid[0] = np.ogrid[np.log10(rangea.min()):np.log10(oldLF.max()):interpolatingSize*1j] grid[1] = np.ogrid[0:6:interpolatingSize*1j] newLF = np.array([[0.for i in range(interpolatingSize)] for i in range(interpolatingSize)]) for i in range(interpolatingSize): for j in range(interpolatingSize): newLF[i][j]= optimize.brentq(rootfinding,1e30,1e50,args=(luminosityParameterHa(grid[1][i]),mpgrid[j])) newLum = interpolate.interpn([grid[0],grid[1]],newLF,np.array([np.log10(oldLF),z[it]]).T,bounds_error=False,fill_value=0.0) newLum = np.power(10,newLum) return newLum
def calculate_convergence(gcf: GeneralizedContinuedFraction,
reference,
plot=False,
title=''):
"""
calculate convergence rate of General Continued Fraction (in reference to some constant x).
the result is the average number of decimal digits, per term of the general continued fraction.
:param plot: whether or not to plot the graph of the log10 difference between convergent and x.
:param gcf: General Continued Fraction to calculate.
:param title: (optional) title of graph.
:param reference: x
"""
q_ = [0, 1]
p_ = [1, gcf.a_[0]]
log_diff = [mpmath.log10(abs((dec(p_[1]) / dec(q_[1])) - reference))]
length = min(200, len(gcf.b_))
for i in range(1, length):
q_.append(gcf.a_[i] * q_[i] + gcf.b_[i - 1] * q_[i - 1])
p_.append(gcf.a_[i] * p_[i] + gcf.b_[i - 1] * p_[i - 1])
if q_[i + 1] == 0:
part_convergent = 0
else:
part_convergent = dec(p_[i + 1]) / dec(q_[i + 1])
if not mpmath.isfinite(part_convergent):
length = i
break
log_diff.append(mpmath.log10(abs(part_convergent - reference)))
if plot:
plt.plot(range(length), log_diff)
plt.title(title)
plt.show()
log_slope = 2 * (log_diff[length - 1] - log_diff[length // 2]) / length
return -log_slope
def p(L, n):
C = mp.log10(2)
po = mp.mp.power(10, mp.floor(mp.log10(L)))
C1 = mp.log10(mp.mpf(L) / po)
C2 = mp.log10(mp.mpf(L + 1) / po)
k = 1
while not (C1 <= mp.frac(k * C) < C2):
k += 1
auxi = lambda k: mp.frac(k * C) - C1
approximators = add_approximators(1000, C)
curr = (k, auxi(k))
treshold = mp.log10(mp.mpf(L + 1) / mp.mpf(L))
idx = 1
while idx < n:
for el in approximators:
if 0 <= curr[1] + el[1] < treshold:
curr = (curr[0] + el[0], auxi(curr[0] + el[0]))
idx += 1
#print(curr[0], idx)
break
return curr[0]
def __compute_surprise_weighted(self, mi, pi, m, p):
b1 = mpmath.binomial(pi, mi)
b2 = mpmath.binomial(p - pi, m - mi)
b3 = mpmath.binomial(p, m)
h3f2 = mpmath.hyp3f2(1, mi - m, mi - pi, mi + 1, -m + mi + p - pi + 1,
1)
#h3f2 = hyper1([mpf(1),mi-m, mi-pi], [mpf(1),mpf(1)+mi, mi+p-pi-m+mpf(1)], 10, 10)
log10cdf = mpmath.log10(b1) + mpmath.log10(b2) + mpmath.log10(
h3f2) - mpmath.log10(b3)
return -float(log10cdf)
def __call__(self, event):
if event.button == 2:
new_x_lim = self.axes.get_xlim()
re_span = self.re_lim[1] - self.re_lim[0]
new_re_lim = (new_x_lim[0] / res[0] * re_span + self.re_lim[0],
new_x_lim[1] / res[0] * re_span + self.re_lim[0])
new_y_lim = self.axes.get_ylim()
im_span = self.im_lim[1] - self.im_lim[0]
new_im_lim = (new_y_lim[0] / res[1] * im_span + self.im_lim[0],
new_y_lim[1] / res[1] * im_span + self.im_lim[0])
new_re_span = new_re_lim[1] - new_re_lim[0]
new_im_span = new_im_lim[1] - new_im_lim[0]
print(int(mpmath.log10(1/new_re_span))+1, int(mpmath.log10(1/new_im_span))+1)
dps = max([int(mpmath.log10(1/new_re_span))+1, int(mpmath.log10(1/new_im_span))+1]) + int(mpmath.log10(max(res))) + 1
mpmath.mp.dps = dps
print(mpmath.mp)
mset = brot_gen_parallel(new_re_lim, new_im_lim, 1000000)
self.axes.cla()
self.axes.imshow(mset, origin="lower", vmin=-0.1, cmap=colmap, interpolation="bilinear")
self.canvas.draw()
self.re_lim = new_re_lim
self.im_lim = new_im_lim
else:
if event.button == 'up':
scale_factor = 1 / base_factor
self.depth *= base_factor
elif event.button == 'down':
scale_factor = base_factor
self.depth /= base_factor
else:
return
cur_xlim = ax.get_xlim()
cur_ylim = ax.get_ylim()
xdata = event.xdata # get event x location
ydata = event.ydata
xlim = [xdata - (xdata - cur_xlim[0])*scale_factor,
xdata + (cur_xlim[1] - xdata)*scale_factor]
ylim = [ydata - (ydata - cur_ylim[0])*scale_factor,
ydata + (cur_ylim[1] - ydata)*scale_factor]
self.axes.set_xlim(xlim[0], xlim[1])
self.axes.set_ylim(ylim[0], ylim[1])
self.canvas.draw()
def fmin_gd(f,
f_dx=None,
x_0=1,
alpha=0,
error=1e-10,
max_iter=1e+5,
alpha_mul=1,
disp=True):
'''
Uses gradient descent algorithm to find a *univariate* function's minimum
Based on mpmath
returns x where f(x) is minimized
f - function to minimize. must only take in x as a parameter
f_dx - optional, first derivative of f
x_0 - optional, initial value of x
alpha - optional, learning rate
error - optional, acceptable error threshold
max_iter - optional, maximum iterations
alpha_mul - optional, multiplier to heuristically determined alpha
disp - optional, prints out Iterations and Final Step if True
Reference: https://en.wikipedia.org/wiki/Gradient_descent#Python
'''
if f_dx is None: f_dx = lambda x: mp.diff(f, x)
# heuristically determine learning rate
if alpha == 0:
try:
alpha = alpha_mul * mp.power(
10, -2 - int(mp.log10(abs(f(x_0)) / abs(x_0))))
except:
alpha = alpha_mul * mp.power(
10, -2 - int(mp.log10(abs(f(x_0 + 0.1)) / abs((x_0 + 0.1)))))
cur_x = x_0
step = alpha * f_dx(cur_x)
ctr = 0
while abs(step) > abs(error):
step = alpha * f_dx(cur_x)
cur_x -= step
ctr += 1
if ctr >= max_iter:
warnings.warn(
"Gradient Descent exited due to max iterations %i.\nReview alpha or x_0."
% (max_iter))
break
if disp: print('Iterations: %i\nFinal Step Size: %.2e' % (ctr, step))
return cur_x
def __compute_surprise_weighted_by_quadrature(self, mi, pi, m, p):
f = lambda i: mpmath.binomial(
self.pi, i) * mpmath.binomial(self.p - self.pi, self.m - i) / \
mpmath.binomial(self.p, self.m)
# +- 0.5 is because of the continuity correction
return float(
-mpmath.log10(mpmath.quad(f, [self.mi - 0.5, self.m + 0.5])))
def getMPFIntegerAsString( n ):
'''Turns an mpmath mpf integer value into a string for use by lexicographic
operators.'''
if n == 0:
return '0'
else:
return nstr( nint( n ), int( floor( log10( n ) + 10 ) ) )[ : -2 ]
def __call__(self,t):
with mp.extradps(self.prec):
t = mpf(t)
if t == 0:
print "ERROR: Inverse transform can not be calculated for t=0"
return ("Error");
N = 2*self.N
# Initiate the stepsize (mit aktueller Präsision)
h = 2*pi/N
# The for loop is evaluating the Laplace inversion at each point theta i
# which is based on the trapezoidal rule
ans = 0.0
for k in range(self.N):
theta = -pi + (k+0.5)*h
z = self.shift + N/t*(Talbot.c1*theta/tan(Talbot.c2*theta) - Talbot.c3 + Talbot.c4*theta)
dz = N/t * (-Talbot.c1*Talbot.c2*theta/sin(Talbot.c2*theta)**2 + Talbot.c1/tan(Talbot.c2*theta)+Talbot.c4)
v1 = exp(z*t)*dz
prec = floor(max(log10(abs(v1)),0))
with mp.extradps(prec):
value = self.F(z)
ans += v1*value
return ((h/pi)*ans).imag
def sig_figs_jy(jn, jn1, yn, yn1, z):
"""Check relation http://dlmf.nist.gov/10.50 .
Parameters
----------
jn : mpf
The value of j_n(x).
jn1 : mpf
The value of j_{n + 1}(x).
yn : mpf
The value of y_n(x).
yn1 : mpf
The value of y_{n + 1}(x).
Returns
-------
The estimated number of significant digits to which the computation of
the passed Bessel functions is correct.
"""
w = mpmath.fabs(z**2 * (jn1 * yn - jn * yn1) - 1)
if not mpmath.isfinite(w):
return w
if w > 0:
return 1 - mpmath.log10(w)
else:
return mpmath.mp.dps
def sig_figs_jy(jn, jn1, yn, yn1, z):
"""Check relation http://dlmf.nist.gov/10.50 .
Parameters
----------
jn : mpf
The value of j_n(x).
jn1 : mpf
The value of j_{n + 1}(x).
yn : mpf
The value of y_n(x).
yn1 : mpf
The value of y_{n + 1}(x).
Returns
-------
The estimated number of significant digits to which the computation of
the passed Bessel functions is correct.
"""
w = mpmath.fabs(z**2*(jn1*yn - jn*yn1) - 1)
if not mpmath.isfinite(w):
return w
if w > 0:
return 1 - mpmath.log10(w)
else:
return mpmath.mp.dps
def single_segment_expect_pv(expect, raw=True):
"""
Compute p-value for an expect value of
a single high scoring segment.
Prob(E >= x) ~ 1 - exp(-E)
This function is equivalent to
single_segment_score_pv
as long as identical units (log)
are used to compute scores and expect
:param expect:
:param raw: return raw P-value instead of -log10(pv)
:return:
"""
with mpm.workprec(NUM_PREC_KA_PV):
x = mpm.convert(expect)
complement = mpm.convert('1')
res = mpm.fsub(complement, mpm.exp(mpm.fneg(x)))
if not raw:
res = mpm.fneg(mpm.log10(res))
res = float(res)
return res
def single_segment_score_pv(score, raw=True):
"""
Compute p-value for normalized local score of
a single high scoring segment.
Computes formula [1] in Karlin & Altschul, PNAS 1993
Prob(S' >= x) ~ 1 - exp(-exp(-x))
:param score:
:param raw: return raw P-value instead of -log10(pv)
:return:
"""
with mpm.workprec(NUM_PREC_KA_PV):
x = mpm.convert(score)
complement = mpm.convert('1')
exponent = mpm.fneg(mpm.exp(mpm.fneg(x)))
res = mpm.fsub(complement, mpm.exp(exponent))
if not raw:
res = mpm.fneg(mpm.log10(res))
res = float(res)
# Equivalent implementation using Python standard library:
#
# x = score
# res = 1 - math.exp(-math.exp(-x))
# if not raw:
# res = -1 * math.log10(res)
return res
def getMPFIntegerAsString(n):
'''Turns an mpmath mpf integer value into a string for use by lexicographic
operators.'''
if n == 0:
return '0'
return nstr(nint(n), int(floor(log10(n) + 10)))[:-2]
def relative_directivity_db(angle,k_v,R_v,alpha_v,An):
'''Calculates on-axis and off-axis angles to get the directivity
for piston in a sphere.
'''
off_axis = d_theta(angle,k_v,R_v,alpha_v,An)
on_axis = d_zero(k_v,R_v,alpha_v,An)
rel_level = 20*mpmath.log10(np.abs(off_axis/on_axis))
return rel_level
def runYAFU(n):
fullOut = subprocess.run([
g.userConfiguration['yafu_path'] + os.sep +
g.userConfiguration['yafu_binary'], '-xover', '120'
],
input='factor(' + str(int(n)) + ')\n',
encoding='ascii',
stdout=subprocess.PIPE,
cwd=g.userConfiguration['yafu_path'],
check=True).stdout
#print( 'out', fullOut )
out = fullOut[fullOut.find('***factors found***'):]
if len(out) < 2:
if log10(n) > 40:
raise ValueError('yafu seems to have crashed trying to factor ' +
str(int(n)) + '.\n\nyafu output follows:\n' +
fullOut)
debugPrint(
'yafu seems to have crashed, switching to built-in factoring code')
return factorise(int(n))
result = []
while True:
prime = ''
out = out[out.find('P'):]
if len(out) < 2:
break
out = out[out.find('='):]
index = 2
while out[index] >= '0' and out[index] <= '9':
prime += out[index]
index += 1
result.append(mpmathify(prime))
if not result:
raise ValueError('yafu seems to have failed.')
answer = fprod(result)
if answer != n:
debugPrint('\nyafu has barfed')
for i in result:
n = fdiv(n, i)
result.extend(runYAFU(n))
return sorted(result)
def getECMFactors( target ):
from pyecm import factors
n = int( floor( target ) )
verbose = g.verbose
randomSigma = True
asymptoticSpeed = 10
processingPower = 1.0
if n < -1:
return [ ( -1, 1 ) ] + getECMFactors( fneg( n ) )
elif n == -1:
return [ ( -1, 1 ) ]
elif n == 0:
return [ ( 0, 1 ) ]
elif n == 1:
return [ ( 1, 1 ) ]
if verbose:
print( '\nfactoring', n, '(', int( floor( log10( n ) ) ), ' digits)...' )
if g.factorCache is None:
loadFactorCache( )
if n in g.factorCache:
if verbose and n != 1:
print( 'cache hit:', n )
print( )
return g.factorCache[ n ]
result = [ ]
for factor in factors( n, verbose, randomSigma, asymptoticSpeed, processingPower ):
result.append( factor )
result = [ int( i ) for i in result ]
largeFactors = list( collections.Counter( [ i for i in result if i > 65535 ] ).items( ) )
product = int( fprod( [ power( i[ 0 ], i[ 1 ] ) for i in largeFactors ] ) )
save = False
if product not in g.factorCache:
g.factorCache[ product ] = largeFactors
save = True
result = list( collections.Counter( result ).items( ) )
if n > g.minValueToCache and n not in g.factorCache:
g.factorCache[ n ] = result
g.factorCacheIsDirty = True
if verbose:
print( )
return result
def addDigits(n, k):
digits = int(k)
if digits == 0:
digitCount = 1
else:
digitCount = int(fadd(floor(log10(k)), 1))
return appendDigits(n, digits, digitCount)
def get_ratios(self):
if self._ratio == None:
self.check()
self._ratio = []
for i in range(len(self._second_distances)):
if self._first_distances[i] != None and self._second_distances[
i] != None:
if self._second_distances[i] >= self._first_distances[i]:
self._ratio.append(
mpmath.log10(self._second_distances[i] /
self._first_distances[i]))
else:
self._ratio.append(
-mpmath.log10(self._first_distances[i] /
self._second_distances[i]))
return self._ratio
def relative_directivity_db(angle, k_v, R_v, alpha_v):
'''
Calculates the :math:`D_{\\theta}/D_{0}` in dB for all angles
in the cap in a sphere model.
'''
off_axis = d_theta(k_v, R_v, alpha_v, angle)
on_axis = d_zero(k_v, R_v, alpha_v)
rel_level = 20*mpmath.log10(abs(off_axis/on_axis))
return rel_level
def test(self,do_twoy=False,up_to_M=0):
r""" Return the number of digits we believe are correct (at least)
EXAMPLES::
sage: G=MySubgroup(Gamma0(1))
sage: R=mpmath.mpf(9.53369526135355755434423523592877032382125639510725198237579046413534)
sage: F=MaassWaveformElement(G,R)
sage: F.test()
7
"""
# If we have a Gamma_0(N) we can use Hecke operators
if(not do_twoy and (isinstance(self._space,sage.modular.arithgroup.congroup_gamma0.Gamma0_class) or \
self._space._G.is_congruence)):
# isinstance(self._space,sage.modular.arithgroup.congroup_sl2z.SL2Z_class_with_category))):
if(self._space._verbose>1):
print "Check Hecke relations!"
er=test_Hecke_relations(2,3,self.coeffs)
d=floor(-mpmath.log10(er))
if(self._space._verbose>1):
print "Hecke is ok up to ",d,"digits!"
return d
else:
# Test two Y's
nd=self._nd+5
[M0,Y0]=find_Y_and_M(G,R,nd)
Y1=Y0*0.95
C1=Maassform_coeffs(self,R,Mset=M0,Yset=Y0 ,ndigs=nd )[0]
C2=Maassform_coeffs(self,R,Mset=M0,Yset=Y1 ,ndigs=nd )[0]
er=mpmath.mpf(0)
for j in range(2,max(M0/2,up_to_M)):
t=abs(C1[j]-C2[j])
print "|C1-C2[",j,"]|=|",C1[j],'-',C2[j],'|=',t
if(t>er):
er=t
d=floor(-mpmath.log10(er))
print "Hecke is ok up to ",d,"digits!"
return d
def compareLists(result1, result2):
if len(result1) != len(result2):
raise ValueError('lists are not of equal length:', len(result1),
len(result2))
for i in range(0, len(result1)):
if isinstance(result1[i], RPNGenerator):
return compareLists(list(result1[i].getGenerator()), result2[i])
if isinstance(result2[i], RPNGenerator):
return compareResults(result1[i], list(result2[i].getGenerator()))
if isinstance(result1[i], list) != isinstance(result2[i], list):
raise ValueError('lists are nested to different levels:', result1,
result2)
if isinstance(result1[i], list) and isinstance(result2[i], list):
compareLists(result1[i], result2[i])
else:
if isinf(result1[i]):
if isinf(result2[i]):
return True
else:
print('**** error in results comparison')
print(type(result1[i]), type(result2[i]))
print(result1[i], result2[i], 'are not equal')
raise ValueError('unit test failed')
if not compareValues(result1[i], result2[i]):
digits = max(log10(result1[i]), log10(result2[i])) + 5
mp.dps = digits
print('**** error in results comparison')
print(type(result1[i]), type(result2[i]))
print(result1[i], result2[i], 'are not equal')
print('difference', fsub(result1[i], result2[i]))
print('difference found at index', i)
raise ValueError('unit test failed')
return True
def G(self,s): # Laplace-Transform
zz = 2*self.v + 2 + s
mu = sqrt(self.v**2+2*zz)
a = mu/2 - self.v/2 - 1
b = mu/2 + self.v/2 + 2
v1 = (2*self.alp)**(-a)*gamma(b)/gamma(mu+1)/(zz*(zz - 2*(1 + self.v)))
prec = floor(max(log10(abs(v1)),mp.dps))+self.prec
# additional precision needed for computation of hyp1f1
with mp.extradps(prec):
value = hyp1f1(a,mu + 1,self.beta)*v1
return value
def getFactors(target):
if target < -1:
result = [-1]
result.extend(getFactors(fneg(target)))
return result
if target == -1:
return [-1]
if target == 0:
return [0]
if target == 1:
return [1]
n = int(floor(target))
setAccuracy(floor(fadd(log10(n), 2)))
if g.factorCache is None:
loadFactorCache()
if not g.ignoreCache:
if n in g.factorCache:
if g.verbose and n != 1:
print('cache hit:', n)
print()
debugPrint('\nfactor cache', n, g.factorCache[n])
return g.factorCache[n]
try:
result = factorByTrialDivision(n) # throws if n is too big
if n > g.minValueToCache and n not in g.factorCache:
g.factorCache[n] = result
return result
except ValueError:
pass
if g.useYAFU and n > g.minValueForYAFU:
result = runYAFU(n)
else:
result = factorise(int(n))
if n > g.minValueToCache:
if g.ignoreCache or (not g.ignoreCache and n not in g.factorCache):
debugPrint('\ncaching factors of ', n, result)
g.factorCache[n] = result
return result
def brute_force(N):
'''
Just to test the actual function
'''
k = 1
idx = 0
for i in range(N):
if str(k)[:3] == str(L):
idx += 1
print(mp.nstr(mp.log10(k) / C), idx)
k *= 2
def publish_drawing_rect(self):
print("------*----- publish zoom")
if self._qim is None:
return
# print("publish", self._rect_pos0, self._rect_pos1)
# print("fractal", self._presenter["fractal"])
# print("fractal", self._presenter["image"])
nx = self._fractal_zoom_init["nx"]
ny = self._fractal_zoom_init["ny"]
# new center offet in pixel
center_off_px = 0.5 * (self._rect_pos0.x() + self._rect_pos1.x() - nx)
center_off_py = 0.5 * (ny - self._rect_pos0.y() - self._rect_pos1.y())
dx_pix = abs(self._rect_pos0.x() - self._rect_pos1.x())
# print("center px", center_off_px)
# print("center py", center_off_py)
ref_zoom = self._fractal_zoom_init.copy()
# str -> mpf as needed
to_mpf = {k: isinstance(self._fractal_zoom_init[k], str) for k in
["x", "y", "dx"]}
# We may need to increase the dps to hold sufficent digits
if to_mpf["dx"]:
ref_zoom["dx"] = mpmath.mpf(ref_zoom["dx"])
pix = ref_zoom["dx"] / float(ref_zoom["nx"])
with mpmath.workdps(6):
# Sets the working dps to 10e-8 x pixel size
ref_zoom["dps"] = int(-mpmath.log10(pix * dx_pix / nx) + 8)
print("------*----- NEW dps from zoom", ref_zoom["dps"])
# if ref_zoom["dps"] > mpmath.dps:
# zoom_dps = max(ref_zoom["dps"], mpmath.mp.dps)
with mpmath.workdps(ref_zoom["dps"]):
for k in ["x", "y"]:
if to_mpf[k]:
ref_zoom[k] = mpmath.mpf(ref_zoom[k])
# print("is_mpf", to_mpf, ref_zoom)
ref_zoom["x"] += center_off_px * pix
ref_zoom["y"] += center_off_py * pix
ref_zoom["dx"] = dx_pix * pix
# mpf -> str (back)
for (k, v) in to_mpf.items():
if v:
if k == "dx":
ref_zoom[k] = mpmath.nstr(ref_zoom[k], 16)
else:
ref_zoom[k] = str(ref_zoom[k])
for key in ["x", "y", "dx", "dps"]:
self._presenter[key] = ref_zoom[key]
def runYAFU( n ):
import subprocess
full_out = subprocess.run( [ g.userConfiguration[ 'yafu_path' ] + os.sep +
g.userConfiguration[ 'yafu_binary' ], str( int( n ) ), '-xover', '120' ],
stdout=subprocess.PIPE, cwd=g.userConfiguration[ 'yafu_path' ] ).stdout.decode( 'ascii' )
#print( 'out', full_out )
out = full_out[ full_out.find( '***factors found***' ) : ]
if len( out ) < 2:
if log10( n ) > 40:
raise ValueError( 'yafu seems to have crashed trying to factor ' + str( int( n ) ) +
'.\n\nyafu output follows:\n' + full_out )
else:
debugPrint( 'yafu seems to have crashed, switching to built-in factoring code' )
from rpn.factorise import factorise
return factorise( int( n ) )
result = [ ]
while True:
prime = ''
out = out[ out.find( 'P' ) : ]
if len( out ) < 2:
break
out = out[ out.find( '=' ) : ]
index = 2
while out[ index ] >= '0' and out[ index ] <= '9':
prime += out[ index ]
index += 1
result.append( mpmathify( prime ) )
if not result:
raise ValueError( 'yafu seems to have failed.' )
answer = fprod( result )
if answer != n:
debugPrint( '\nyafu has barfed' )
for i in result:
n = fdiv( n, i )
result.extend( runYAFU( n ) )
return sorted( result )
def addDigits( n, k ):
if real( k ) < 0:
raise ValueError( "'add_digits' requires a non-negative integer for the second argument" )
digits = int( k )
if digits == 0:
digitCount = 1
else:
digitCount = int( fadd( floor( log10( k ) ), 1 ) )
return appendDigits( n, digits, digitCount )
def log10sumexp(self, log10_array):
'''
return value approxiameting to log10(sum(10^log10_probs))
'''
if (self.speedUp):
n_array = self.ffi.cast("int", len(log10_array))
log10_array_c = self.ffi.new("double []", log10_array)
return self.lib.log10sumexp(log10_array_c, n_array)
m = max(log10_array)
return m + math.log10(sum(pow(10.0, x - m) for x in log10_array))
def multi_segment_score_pv(score, num_segments, raw=True):
"""
Compute p-value for normalized score when considering
multiple high scoring segments. This functions considers
the normalized score Sr', i.e., the normalized score
of the HSP at rank r. For r=1, this formula is equivalent
to single_segment_score_pv
Computes formula [3] in Karlin & Altschul, PNAS 1993
Prob(Sr' >= x) ~ 1 - exp(-exp(-x)) * SUM (k=0 ... r - 1) { exp(-kx) / k! }
Implementation detail:
Python's range is not right-inclusive,
go up to r, not r - 1 for summation
:param score:
:param num_segments:
:param raw: return raw P-value instead of -log10(pv)
:return:
"""
with mpm.workprec(NUM_PREC_KA_PV):
def create_summand(sum_x, k):
prec_k = mpm.convert(k)
enum = mpm.exp(mpm.fneg(mpm.fmul(prec_k, sum_x)))
denom = mpm.factorial(prec_k)
summand = mpm.fdiv(enum, denom)
return summand
x = mpm.convert(score)
r = num_segments
complement = mpm.convert('1')
factor1 = mpm.exp(mpm.fneg(mpm.exp(mpm.fneg(x))))
factor2 = mpm.fsum(map(lambda k: create_summand(x, k), range(0, r)))
res = mpm.fsub(complement, mpm.fmul(factor1, factor2))
if not raw:
res = mpm.fneg(mpm.log10(res))
res = float(res)
# Equivalent implementation using Python standard library:
#
# x = score
# r = num_segments
# factor_1 = math.exp(-math.exp(-x))
# factor_2 = math.fsum(map(lambda k: math.exp(-k * x) / math.factorial(k), range(0, r)))
# res = 1 - factor_1 * factor_2
# if not raw:
# res = -1 * math.log10(res)
return res
def getPartitionNumber( n ):
'''
This version is, um, less recursive than the original, which I've kept.
The strategy is to create a list of the smaller partition numbers we need
to calculate and then start calling them recursively, starting with the
smallest. This will minimize the number of recursions necessary, and in
combination with caching values, will calculate practically any integer
partition without the risk of a stack overflow.
I can't help but think this is still grossly inefficient compared to what's
possible. It seems that using this algorithm, calculating any integer
partition ends up necessitating calculating the integer partitions of
every integer smaller than the original argument.
'''
debugPrint( 'partition', int( n ) )
if real_int( n ) < 0:
raise ValueError( 'non-negative argument expected' )
elif n in ( 0, 1 ):
return 1
sign = 1
i = 1
k = 1
estimate = log10( fdiv( power( e, fmul( pi, sqrt( fdiv( fmul( 2, n ), 3 ) ) ) ),
fprod( [ 4, n, sqrt( 3 ) ] ) ) )
if mp.dps < estimate + 5:
mp.dps = estimate + 5
partitionList = [ ]
signList = [ ]
while n - k >= 0:
partitionList.append( ( fsub( n, k ), sign ) )
i += 1
if i % 2:
sign *= -1
k = getNthGeneralizedPolygonalNumber( i, 5 )
partitionList = partitionList[ : : -1 ]
total = 0
for partition, sign in partitionList:
total = fadd( total, fmul( sign, getPartitionNumber( partition ) ) )
return total
def isKMorphic(n, k):
'''
Returns true if n to the k power ends with n.
This code won't work correctly for integral powers of 10, but they can
never be morphic anyway, except for 1, which I handle specially.
'''
if n == 1:
return 1
modulo = power(10, ceil(log10(n)))
powmod = getPowMod(n, k, modulo)
return 1 if (n == powmod) else 0
def evaluate(algorithm: str, accuracy: int) -> mpmath.mpf:
"""Retrun Logarithm of the error between calculated pi and actural one.
Args:
algorithm (str): algorithm by which pi is calcurated
accuracy (int): accuracy of calculated pi
Returns:
mpmath.mpf: logarithms or error of calculated pi and actual one
"""
actual_pi: mpmath.mpf = calc('actual', ACTUAL_PI_DIGIT)
compared_pi: mpmath.mpf = calc(algorithm, accuracy)
subtraction: mpmath.mpf = mpmath.fsub(actual_pi, compared_pi)
mpmath.mp.dps = ERROR_ACCRACY
return -mpmath.log10(subtraction)
def getPartitionNumber(n):
'''
This version is, um, less recursive than the original, which I've kept.
The strategy is to create a list of the smaller partition numbers we need
to calculate and then start calling them recursively, starting with the
smallest. This will minimize the number of recursions necessary, and in
combination with caching values, will calculate practically any integer
partition without the risk of a stack overflow.
I can't help but think this is still grossly inefficient compared to what's
possible. It seems that using this algorithm, calculating any integer
partition ends up necessitating calculating the integer partitions of
every integer smaller than the original argument.
'''
debugPrint('partition', int(n))
if n in (0, 1):
return 1
sign = 1
i = 1
k = 1
estimate = log10(
fdiv(power(e, fmul(pi, sqrt(fdiv(fmul(2, n), 3)))),
fprod([4, n, sqrt(3)])))
if mp.dps < estimate + 5:
mp.dps = estimate + 5
partitionList = []
while n - k >= 0:
partitionList.append((fsub(n, k), sign))
i += 1
if i % 2:
sign *= -1
k = getNthGeneralizedPolygonalNumber(i, 5)
partitionList = partitionList[::-1]
total = 0
for partition, sign in partitionList:
total = fadd(total, fmul(sign, getPartitionNumber(partition)))
return total
def assert_equal(vardict_1, vardict_2, suppress_message=False):
""" Assert SymPy Expression Equality
>>> from sympy import sin, cos
>>> from sympy.abc import x
>>> assert_equal(sin(2*x), 2*sin(x)*cos(x))
Assertion Passed!
>>> assert_equal(cos(2*x), cos(x)**2 - sin(x)**2)
Assertion Passed!
>>> assert_equal(cos(2*x), 1 - 2*sin(x)**2)
Assertion Passed!
>>> assert_equal(cos(2*x), 1 + 2*sin(x)**2)
Traceback (most recent call last):
...
AssertionError
>>> vardict_1 = {'A': sin(2*x), 'B': cos(2*x)}
>>> vardict_2 = {'A': 2*sin(x)*cos(x), 'B': cos(x)**2 - sin(x)**2}
>>> assert_equal(vardict_1, vardict_2)
Assertion Passed!
>>> assert_equal('(a^2 - b^2) - (a + b)*(a - b)', 0)
Assertion Passed!
"""
if not isinstance(vardict_1, dict):
vardict_1 = {'': vardict_1}
if not isinstance(vardict_2, dict):
vardict_2 = {'': vardict_2}
for var_1, var_2 in zip(vardict_1, vardict_2):
if not isinstance(vardict_1[var_1], sp.Basic):
vardict_1[var_1] = sp.sympify(vardict_1[var_1])
if not isinstance(vardict_2[var_2], sp.Basic):
vardict_2[var_2] = sp.sympify(vardict_2[var_2])
# update each vardict with mapping: variable -> (pseudo) unique number
vardict_1, vardict_2 = update_vardict(vardict_1), update_vardict(vardict_2)
# assert whether SDA >= (2/3) * precision, implying expression equality
for var_1, var_2 in zip(vardict_1, vardict_2):
n_1, n_2 = vardict_1[var_1], vardict_2[var_2]
if n_1 == n_2: continue
E_rel = 2 * fabs(n_1 - n_2)/(fabs(n_1) + fabs(n_2))
assert -log10(E_rel) + 1 >= (2.0/3) * precision
if not suppress_message:
print('Assertion Passed!')
def getDecimalDigitList(n, k):
result = []
setAccuracy(k)
digits = floor(log10(n))
if digits < 0:
for _ in arange(fsub(fabs(digits), 1)):
result.append(0)
k = fsub(k, fsub(fabs(digits), 1))
value = fmul(n, power(10, fsub(k, fadd(digits, 1))))
for c in getMPFIntegerAsString(floor(value)):
result.append(int(c))
return result
def apply_2(self, n, evaluation):
'MantissaExponent[n_]'
n_sympy = n.to_sympy()
expr = Expression('MantissaExponent', n)
if isinstance(n.to_python(), complex):
evaluation.message('MantissaExponent', 'realx', n)
return expr
# Handle Input with special cases such as PI and E
if n_sympy.is_constant():
temp_n = Expression('N', n).evaluate(evaluation)
py_n = temp_n.to_python()
else:
return expr
base_exp = int(mpmath.log10(py_n))
exp = (base_exp + 1) if base_exp >= 0 else base_exp
return Expression('List', Expression('Divide', n, (10**exp)), exp)
def apply_2(self, n, evaluation):
"MantissaExponent[n_]"
n_sympy = n.to_sympy()
expr = Expression("MantissaExponent", n)
if isinstance(n.to_python(), complex):
evaluation.message("MantissaExponent", "realx", n)
return expr
# Handle Input with special cases such as PI and E
if n_sympy.is_constant():
temp_n = apply_N(n, evaluation)
py_n = temp_n.to_python()
else:
return expr
base_exp = int(mpmath.log10(py_n))
exp = (base_exp + 1) if base_exp >= 0 else base_exp
return Expression("List", Expression("Divide", n, (10**exp)), exp)
def getLog10( n ):
return log10( n )
def compareResults( result1, result2 ):
'''Compares two RPN expressions to make sure they produce the same result.
Does nothing if the results compare successfully, otherwise raises an
exception.'''
if isinstance( result1, RPNGenerator ):
return compareResults( list( result1.getGenerator( ) ), result2 )
if isinstance( result2, RPNGenerator ):
return compareResults( result1, list( result2.getGenerator( ) ) )
if isinstance( result1, list ) != isinstance( result2, list ):
print( '**** error in results comparison' )
print( ' result 1: ', result1 )
print( ' result 2: ', result2 )
raise ValueError( 'one result is a list, the other isn\'t' )
if isinstance( result1, str ) != isinstance( result2, str ):
print( '**** error in results comparison' )
print( ' result 1: ', result1 )
print( ' result 2: ', result2 )
raise ValueError( 'one result is a string, the other isn\'t' )
if isinstance( result1, str ) and isinstance( result2, str ):
if result1 != result2:
print( '**** error in results comparison' )
print( type( result1 ), type( result2 ) )
print( result1, result2, 'are not equal' )
raise ValueError( 'unit test failed' )
else:
return
if isinstance( result1, RPNMeasurement ) and isinstance( result2, RPNMeasurement ):
if result1 != result2:
print( '**** error in results comparison' )
print( type( result1 ), type( result2 ) )
with workdps( 50 ):
print( result1.value, result1.units, result2.value, result2.units, 'are not equal' )
raise ValueError( 'unit test failed' )
else:
return True
if isinstance( result1, list ) and isinstance( result2, list ):
if len( result1 ) != len( result2 ):
raise ValueError( 'lists are not of equal length:', len( result1 ), len( result2 ) )
for i in range( 0, len( result1 ) ):
if isinf( result1[ i ] ):
if isinf( result2[ i ] ):
return True
else:
print( '**** error in results comparison' )
print( type( result1[ i ] ), type( result2[ i ] ) )
print( result1[ i ], result2[ i ], 'are not equal' )
raise ValueError( 'unit test failed' )
if not compareValues( result1[ i ], result2[ i ] ):
digits = max( log10( result1[ i ] ), log10( result2[ i ] ) ) + 5
mp.dps = digits
print( '**** error in results comparison' )
print( type( result1[ i ] ), type( result2[ i ] ) )
print( result1[ i ], result2[ i ], 'are not equal' )
print( 'difference', fsub( result1[ i ], result2[ i ] ) )
print( 'difference found at index', i )
raise ValueError( 'unit test failed' )
else:
if not compareValues( result1, result2 ):
print( '**** error in results comparison' )
print( ' result 1: ', result1 )
print( ' result 2: ', result2 )
raise ValueError( 'unit test failed' )
def getMPFIntegerAsString( n ):
if n == 0:
return '0'
else:
return nstr( nint( n ), int( floor( log10( n ) + 10 ) ) )[ : -2 ]
( 'reaumur', 'delisle' ) : lambda re: fmul( fsub( 80, re ), fdiv( 15, 8 ) ),
( 'reaumur', 'fahrenheit' ) : lambda re: fadd( fmul( re, fdiv( 9, 4 ) ), 32 ),
( 'reaumur', 'kelvin' ) : lambda re: fadd( fmul( re, fdiv( 5, 4 ) ), mpf( '273.15' ) ),
( 'reaumur', 'rankine' ) : lambda re: fadd( fmul( re, fdiv( 9, 4 ) ), mpf( '491.67' ) ),
( 'reaumur', 'romer' ) : lambda re: fadd( fmul( re, fdiv( 21, 32 ) ), mpf( 7.5 ) ),
( 'romer', 'celsius' ) : lambda ro: fmul( fsub( ro, mpf( '7.5' ) ), fdiv( 40, 21 ) ),
( 'romer', 'degree_newton' ) : lambda ro: fmul( fsub( ro, mpf( '7.5' ) ), fdiv( 22, 35 ) ),
( 'romer', 'delisle' ) : lambda ro: fmul( fsub( 60, ro ), fdiv( 20, 7 ) ),
( 'romer', 'fahrenheit' ) : lambda ro: fadd( fmul( fsub( ro, mpf( '7.5' ) ), fdiv( 24, 7 ) ), 32 ),
( 'romer', 'kelvin' ) : lambda ro: fadd( fmul( fsub( ro, mpf( '7.5' ) ), fdiv( 40, 21 ) ), mpf( '273.15' ) ),
( 'romer', 'rankine' ) : lambda ro: fadd( fmul( fsub( ro, mpf( '7.5' ) ), fdiv( 24, 7 ) ), mpf( '491.67' ) ),
( 'romer', 'reaumur' ) : lambda ro: fmul( fsub( ro, mpf( '7.5' ) ), fdiv( 32, 21 ) ),
( 'decibel-volt', 'volt' ) : lambda dBV: power( 10, fdiv( dBV, 10 ) ),
( 'volt', 'decibel-volt' ) : lambda V: fmul( log10( V ), 10 ),
( 'decibel-watt', 'watt' ) : lambda dBW: power( 10, fdiv( dBW, 10 ) ),
( 'watt', 'decibel-watt' ) : lambda W: fmul( log10( W ), 10 ),
( 'decibel-milliwatt', 'watt' ) : lambda dBm: power( 10, fdiv( fsub( dBm, 30 ), 10 ) ),
( 'watt', 'decibel-milliwatt' ) : lambda W: fmul( log10( fmul( W, 1000 ) ), 10 ),
( 'second', 'hertz' ) : lambda second: fdiv( 1, second ),
( 'hertz', 'second' ) : lambda hertz: fdiv( 1, hertz ),
( 'ohm', 'siemens' ) : lambda ohm: fdiv( 1, ohm ),
( 'siemens', 'ohm' ) : lambda siemens: fdiv( 1, siemens ),
}
def describeInteger( n ):
if n < 1:
raise ValueError( "'describe' requires a positive integer argument" )
indent = ' ' * 4
print( )
print( real_int( n ), 'is:' )
if isOdd( n ):
print( indent + 'odd' )
elif isEven( n ):
print( indent + 'even' )
if isPrimeNumber( n ):
isPrime = True
print( indent + 'prime' )
elif n > 3:
isPrime = False
print( indent + 'composite' )
else:
isPrime = False
if isKthPower( n, 2 ):
print( indent + 'the ' + getShortOrdinalName( sqrt( n ) ) + ' square number' )
if isKthPower( n, 3 ):
print( indent + 'the ' + getShortOrdinalName( cbrt( n ) ) + ' cube number' )
for i in arange( 4, fadd( ceil( log( fabs( n ), 2 ) ), 1 ) ):
if isKthPower( n, i ):
print( indent + 'the ' + getShortOrdinalName( root( n, i ) ) + ' ' + \
getNumberName( i, True ) + ' power' )
# triangular
guess = findPolygonalNumber( n, 3 )
if getNthPolygonalNumber( guess, 3 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' triangular number' )
# pentagonal
guess = findPolygonalNumber( n, 5 )
if getNthPolygonalNumber( guess, 5 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' pentagonal number' )
# hexagonal
guess = findPolygonalNumber( n, 6 )
if getNthPolygonalNumber( guess, 6 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' hexagonal number' )
# heptagonal
guess = findPolygonalNumber( n, 7 )
if getNthPolygonalNumber( guess, 7 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' heptagonal number' )
# octagonal
guess = findPolygonalNumber( n, 8 )
if getNthPolygonalNumber( guess, 8 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' octagonal number' )
# nonagonal
guess = findPolygonalNumber( n, 9 )
if getNthPolygonalNumber( guess, 9 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' nonagonal number' )
# decagonal
guess = findPolygonalNumber( n, 10 )
if getNthPolygonalNumber( guess, 10 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' decagonal number' )
#if n > 1:
# for i in range( 11, 101 ):
# if getNthPolygonalNumber( findPolygonalNumber( n, i ), i ) == n:
# print( indent + str( i ) + '-gonal' )
# centered triangular
guess = findCenteredPolygonalNumber( n, 3 )
if getNthCenteredPolygonalNumber( guess, 3 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered triangular' )
# centered square
guess = findCenteredPolygonalNumber( n, 4 )
if getNthCenteredPolygonalNumber( guess, 4 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered square number' )
# centered pentagonal
guess = findCenteredPolygonalNumber( n, 5 )
if getNthCenteredPolygonalNumber( guess, 5 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered pentagonal number' )
# centered hexagonal
guess = findCenteredPolygonalNumber( n, 6 )
if getNthCenteredPolygonalNumber( guess, 6 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered hexagonal number' )
# centered heptagonal
guess = findCenteredPolygonalNumber( n, 7 )
if getNthCenteredPolygonalNumber( guess, 7 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered heptagonal number' )
# centered octagonal
guess = findCenteredPolygonalNumber( n, 8 )
if getNthCenteredPolygonalNumber( guess, 8 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered octagonal number' )
# centered nonagonal
guess = findCenteredPolygonalNumber( n, 9 )
if getNthCenteredPolygonalNumber( guess, 9 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered nonagonal number' )
# centered decagonal
guess - findCenteredPolygonalNumber( n, 10 )
if getNthCenteredPolygonalNumber( guess, 10 ) == n:
print( indent + 'the ' + getShortOrdinalName( guess ) + ' centered decagonal number' )
# pandigital
if isPandigital( n ):
print( indent + 'pandigital' )
#for i in range( 4, 21 ):
# if isBaseKPandigital( n, i ):
# print( indent + 'base ' + str( i ) + ' pandigital' )
# Fibonacci
result = findInput( n, fib, lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Fibonacci number' )
# Tribonacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 3 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Tribonacci number' )
# Tetranacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 4 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Tetranacci number' )
# Pentanacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 5 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Pentanacci number' )
# Hexanacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 6 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Hexanacci number' )
# Heptanacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 7 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Heptanacci number' )
# Octanacci
result = findInput( n, lambda n : getNthKFibonacciNumber( n, 8 ), lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Octanacci number' )
# Lucas numbers
result = findInput( n, getNthLucasNumber, lambda n: fmul( log10( n ), 5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Lucas number' )
# base-k repunits
if n > 1:
for i in range( 2, 21 ):
result = findInput( n, lambda x: getNthBaseKRepunit( x, i ), lambda n: log( n, i ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' base-' + str( i ) + ' repunit' )
# Jacobsthal numbers
result = findInput( n, getNthJacobsthalNumber, lambda n: fmul( log( n ), 1.6 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Jacobsthal number' )
# Padovan numbers
result = findInput( n, getNthPadovanNumber, lambda n: fmul( log10( n ), 9 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Padovan number' )
# Fibonorial numbers
result = findInput( n, getNthFibonorial, lambda n: sqrt( fmul( log( n ), 10 ) ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Fibonorial number' )
# Mersenne primes
result = findInput( n, getNthMersennePrime, lambda n: fadd( fmul( log( log( sqrt( n ) ) ), 2.7 ), 3 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Mersenne prime' )
## perfect number
result = findInput( n, getNthPerfectNumber, lambda n: fmul( log( log( n ) ), 2.6 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' perfect number' )
# Mersenne exponent
result = findInput( n, getNthMersenneExponent, lambda n: fmul( log( n ), 2.7 ), max=50 )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Mersenne exponent' )
if not isPrime and n != 1 and n <= largestNumberToFactor:
# deficient
if isDeficient( n ):
print( indent + 'deficient' )
# abundant
if isAbundant( n ):
print( indent + 'abundant' )
# k_hyperperfect
for i in sorted( list( set( sorted( downloadOEISSequence( 34898 )[ : 500 ] ) ) ) )[ 1 : ]:
if i > n:
break
if isKHyperperfect( n, i ):
print( indent + str( i ) + '-hyperperfect' )
break
# smooth
for i in getPrimes( 2, 50 ):
if isSmooth( n, i ):
print( indent + str( i ) + '-smooth' )
break
# rough
previousPrime = 2
for i in getPrimes( 2, 50 ):
if not isRough( n, i ):
print( indent + str( previousPrime ) + '-rough' )
break
previousPrime = i
# is_semiprime
if isSemiprime( n ):
print( indent + 'semiprime' )
# is_sphenic
elif isSphenic( n ):
print( indent + 'sphenic' )
elif isSquareFree( n ):
print( indent + 'square-free' )
# powerful
if isPowerful( n ):
print( indent + 'powerful' )
# factorial
result = findInput( n, getNthFactorial, lambda n: power( log10( n ), 0.92 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' factorial number' )
# alternating factorial
result = findInput( n, getNthAlternatingFactorial, lambda n: fmul( sqrt( log( n ) ), 0.72 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' alternating factorial number' )
# double factorial
if n == 1:
result = ( True, 1 )
else:
result = findInput( n, getNthDoubleFactorial, lambda n: fdiv( power( log( log( n ) ), 4 ), 7 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' double factorial number' )
# hyperfactorial
result = findInput( n, getNthHyperfactorial, lambda n: fmul( sqrt( log( n ) ), 0.8 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' hyperfactorial number' )
# subfactorial
result = findInput( n, getNthSubfactorial, lambda n: fmul( log10( n ), 1.1 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' subfactorial number' )
# superfactorial
if n == 1:
result = ( True, 1 )
else:
result = findInput( n, getNthSuperfactorial, lambda n: fadd( sqrt( log( n ) ), 1 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' superfactorial number' )
# pernicious
if isPernicious( n ):
print( indent + 'pernicious' )
# pronic
if isPronic( n ):
print( indent + 'pronic' )
# Achilles
if isAchillesNumber( n ):
print( indent + 'an Achilles number' )
# antiharmonic
if isAntiharmonic( n ):
print( indent + 'antiharmonic' )
# unusual
if isUnusual( n ):
print( indent + 'unusual' )
# hyperperfect
for i in range( 2, 21 ):
if isKHyperperfect( n, i ):
print( indent + str( i ) + '-hyperperfect' )
# Ruth-Aaron
if isRuthAaronNumber( n ):
print( indent + 'a Ruth-Aaron number' )
# Smith numbers
if isSmithNumber( n ):
print( indent + 'a Smith number' )
# base-k Smith numbers
for i in range( 2, 10 ):
if isBaseKSmithNumber( n, i ):
print( indent + 'a base-' + str( i ) + ' Smith number' )
# order-k Smith numbers
for i in range( 2, 11 ):
if isOrderKSmithNumber( n, i ):
print( indent + 'an order-' + str( i ) + ' Smith number' )
# polydivisible
if isPolydivisible( n ):
print( indent + 'polydivisible' )
# Carol numbers
result = findInput( n, getNthCarolNumber, lambda n: fmul( log10( n ), fdiv( 5, 3 ) ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Carol number' )
# Kynea numbers
result = findInput( n, getNthKyneaNumber, lambda n: fmul( log10( n ), fdiv( 5, 3 ) ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Kynea number' )
# Leonardo numbers
result = findInput( n, getNthLeonardoNumber, lambda n: fsub( log( n, phi ), fdiv( 1, phi ) ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Leonardo number' )
# Riesel numbers
result = findInput( n, getNthRieselNumber, lambda n: fmul( log( n ), 1.25 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Riesel number' )
# Thabit numbers
result = findInput( n, getNthThabitNumber, lambda n: fmul( log10( n ), 3.25 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Thabit number' )
# Carmmichael
if isCarmichaelNumber( n ):
print( indent + 'a Carmichael number' )
# narcissistic
if isNarcissistic( n ):
print( indent + 'narcissistic' )
# PDI
if isPerfectDigitalInvariant( n ):
print( indent + 'a perfect digital invariant' )
# PDDI
if isPerfectDigitToDigitInvariant( n, 10 ):
print( indent + 'a perfect digit-to-digit invariant in base 10' )
# Kaprekar
if isKaprekar( n ):
print( indent + 'a Kaprekar number' )
# automorphic
if isAutomorphic( n ):
print( indent + 'automorphic' )
# trimorphic
if isTrimorphic( n ):
print( indent + 'trimorphic' )
# k-morphic
for i in range( 4, 11 ):
if isKMorphic( n, i ):
print( indent + str( i ) + '-morphic' )
# bouncy
if isBouncy( n ):
print( indent + 'bouncy' )
# step number
if isStepNumber( n ):
print( indent + 'a step number' )
# Apery numbers
result = findInput( n, getNthAperyNumber, lambda n: fadd( fdiv( log( n ), 1.5 ), 1 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Apery number' )
# Delannoy numbers
result = findInput( n, getNthDelannoyNumber, lambda n: fmul( log10( n ), 1.35 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Delannoy number' )
# Schroeder numbers
result = findInput( n, getNthSchroederNumber, lambda n: fmul( log10( n ), 1.6 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Schroeder number' )
# Schroeder-Hipparchus numbers
result = findInput( n, getNthSchroederHipparchusNumber, lambda n: fdiv( log10( n ), 1.5 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Schroeder-Hipparchus number' )
# Motzkin numbers
result = findInput( n, getNthMotzkinNumber, lambda n: log( n ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Motzkin number' )
# Pell numbers
result = findInput( n, getNthPellNumber, lambda n: fmul( log( n ), 1.2 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Pell number' )
# Sylvester numbers
if n > 1:
result = findInput( n, getNthSylvesterNumber, lambda n: sqrt( sqrt( log( n ) ) ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' Sylvester number' )
# partition numbers
result = findInput( n, getPartitionNumber, lambda n: power( log( n ), 1.56 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' partition number' )
# menage numbers
result = findInput( n, getNthMenageNumber, lambda n: fdiv( log10( n ), 1.2 ) )
if result[ 0 ]:
print( indent + 'the ' + getShortOrdinalName( result[ 1 ] ) + ' menage number' )
print( )
print( int( n ), 'has:' )
# number of digits
digits = log10( n )
if isInteger( digits ):
digits += 1
else:
digits = ceil( digits )
print( indent + str( int( digits ) ) + ' digit' + ( 's' if digits > 1 else '' ) )
# digit sum
print( indent + 'a digit sum of ' + str( int( sumDigits( n ) ) ) )
# digit product
digitProduct = multiplyDigits( n )
print( indent + 'a digit product of ' + str( int( digitProduct ) ) )
# non-zero digit product
if digitProduct == 0:
print( indent + 'a non-zero digit product of ' + str( int( multiplyNonzeroDigits( n ) ) ) )
if not isPrime and n != 1 and n <= largestNumberToFactor:
# factors
factors = getFactors( n )
factorCount = len( factors )
print( indent + str( factorCount ) + ' prime factor' + ( 's' if factorCount > 1 else '' ) + \
': ' + ', '.join( [ str( int( i ) ) for i in factors ] ) )
# number of divisors
divisorCount = int( getDivisorCount( n ) )
print( indent + str( divisorCount ) + ' divisor' + ( 's' if divisorCount > 1 else '' ) )
if n <= largestNumberToFactor:
print( indent + 'a sum of divisors of ' + str( int( getSigma( n ) ) ) )
print( indent + 'a Stern value of ' + str( int( getNthStern( n ) ) ) )
calkin_wilf = getNthCalkinWilf( n )
print( indent + 'a Calkin-Wilf value of ' + str( int( calkin_wilf[ 0 ] ) ) + '/' + str( int( calkin_wilf[ 1 ] ) ) )
print( indent + 'a Mobius value of ' + str( int( getMobius( n ) ) ) )
print( indent + 'a radical of ' + str( int( getRadical( n ) ) ) )
print( indent + 'a Euler phi value of ' + str( int( getEulerPhi( n ) ) ) )
print( indent + 'a digital root of ' + str( int( getDigitalRoot( n ) ) ) )
if hasUniqueDigits( n ):
print( indent + 'unique digits' )
print( indent + 'a multiplicative persistence of ' + str( int( getPersistence( n ) ) ) )
print( indent + 'an Erdos persistence of ' + str( int( getErdosPersistence( n ) ) ) )
if n > 9 and isIncreasing( n ):
if not isDecreasing( n ):
print( indent + 'increasing digits' )
elif n > 9 and isDecreasing( n ):
print( indent + 'decreasing digits' )
print( )
return n
def LuminosityFunction(luminosityParameter,luminosity): return np.log10(luminosityParameter[0])+float(mpmath.log10(mpmath.gammainc(luminosityParameter[2]+1,luminosity/luminosityParameter[1])))
def rootfinding(luminosity,luminosityParameter,nold): return np.log10(luminosityParameter[0][0])+float(mpmath.log10(mpmath.gammainc(luminosityParameter[2][0]+1,luminosity/luminosityParameter[1][0])))-float(nold)