def OLDgetPartitionNumber( n ):
if n < 0:
return 0
if n < 2:
return 1
result = mpmathify( 0 )
for k in arange( 1, n + 1 ):
#n1 = n - k * ( 3 * k - 1 ) / 2
n1 = fsub( n, fdiv( fmul( k, fsub( fmul( 3, k ), 1 ) ), 2 ) )
#n2 = n - k * ( 3 * k + 1 ) / 2
n2 = fsub( n, fdiv( fmul( k, fadd( fmul( 3, k ), 1 ) ), 2 ) )
result = fadd( result, fmul( power( -1, fadd( k, 1 ) ), fadd( getPartitionNumber( n1 ), getPartitionNumber( n2 ) ) ) )
if n1 <= 0:
break
#old = NOT_QUITE_AS_OLDgetPartitionNumber( n )
#
#if ( old != result ):
# raise ValueError( "It's broke." )
return result
def createStructure(self):
startw = (self.widthSmall / 2) + self.xpoz
startd = (self.depthSmall / 2) + self.ypoz
starth = (self.heightSmall / 2) + self.zpoz
for i in mp.arange(self.wElements):
for j in mp.arange(self.dElements):
for k in mp.arange(self.hElements):
if self.isInStructure(startw+self.widthSmall*i, startd+self.depthSmall*j, starth+self.heightSmall*k):
self.smallBlocksStructure.append([startw+self.widthSmall*i, startd+self.depthSmall*j, starth+self.heightSmall*k])
self.nElements = mp.mpmathify(len(self.smallBlocksStructure))
def tupper_bmp(self, i: mp.mpf = 0) -> np.matrix:
"""
Returns a bitmap representing the i-th shift from the Tupper's original self-referential formula
:param i: i-th shift
:return: boolean matrix representing the bitmap
"""
bmp = mp.matrix(self.height, self.width)
for x in mp.arange(self.height):
for y in mp.arange(self.width):
bmp[x, y] = TupperPy.tupper(x, (self.k + y + 17 * i))
np_bmp = np.matrix(bmp.transpose().tolist(), dtype=bool)
return np.flip(np_bmp, 1)
def test_Diode_In_mpmath():
b = [mpm.mpf('1.238e-14'), mpm.mpf('1.3'), mpm.mpf('1')]
c = None
#0.0026638081177255196
rng = mpm.arange('0.01', '2', '0.01')
xstart = [mpm.mpf('0.00001')]
xend = [mpm.mpf('2')]
Ve = np.array([[0.0001]])
#снимем ВАХ
# resrng=[solver_Diode_In_mpmath ([x],b)[0] for x in rng] # изменяем напряжение на базе при постоянном напряжении на коллекторе - снимаем ток базы.
#resrng=o_pmpm.makeMeasOneDot_lognorm(solver_Diode_In_mpmath, x],b)[0] for x in rng] # изменяем напряжение на базе при постоянном напряжении на коллекторе - снимаем ток базы.
resrng = [
o_pmpm.makeMeasOneDot_lognorm(solver_Diode_In_mpmath, [x], b, c, Ve)
for x in rng
]
resrng1 = [
solver_Diode_In_mpmath([x], b)[0] for x in rng
] # изменяем напряжение на базе при постоянном напряжении на коллекторе - снимаем ток базы.
# plt.plot(rng , resrngorig, label='r=0')
plt.plot(rng, resrng, label='r=1000')
plt.plot(rng, resrng1, label='r=3000')
plt.legend(loc='upper left')
#plt.axis([0.0,1.0,0,5])
plt.grid()
plt.show()
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 getNthPolyPrime( n, poly ):
result = getNthPrime( n )
for i in arange( 1, poly ):
result = getNthPrime( result )
return result
def getNthDelannoyNumber( n ):
result = 0
for k in arange( 0, fadd( real( n ), 1 ) ):
result = fadd( result, fmul( binomial( n, k ), binomial( fadd( n, k ), k ) ) )
return result
def oldGetPartitionNumber(n):
if n < 2:
return 1
result = mpmathify(0)
for k in arange(1, n + 1):
#n1 = n - k * ( 3 * k - 1 ) / 2
sub1 = fsub(n, fdiv(fmul(k, fsub(fmul(3, k), 1)), 2))
#n2 = n - k * ( 3 * k + 1 ) / 2
sub2 = fsub(n, fdiv(fmul(k, fadd(fmul(3, k), 1)), 2))
result = fadd(
result,
fmul(power(-1, fadd(k, 1)),
fadd(getPartitionNumber(sub1), getPartitionNumber(sub2))))
if sub1 <= 0:
break
#old = NOT_QUITE_AS_OLDgetPartitionNumber( n )
#
#if ( old != result ):
# raise ValueError( "It's broke." )
return result
def tetrate( i, j ):
result = i
for x in arange( 1, j ):
result = power( result, i )
return result
def tetrateRight( i, j ):
result = i
for x in arange( 1, j ):
result = power( i, result )
return result
def getNthSchroederHipparchusNumber( n ):
result = 0
for i in arange( n ):
result = fadd( result, fmul( getNarayanaNumber( n, fadd( i, 1 ) ), power( 2, i ) ) )
return result
def getNthFibonorial( n ):
result = 1
for i in arange( 2, real( n ) ):
result = fmul( result, fib( i ) )
return result
def getNextPrimes( p, k, func=getNextPrimeCandidateForAny ):
result = [ ]
for i in arange( 0, k ):
p = getNextPrime( p, func )
result.append( p )
return result
def getNthMotzkinNumber( n ):
result = 0
for j in arange( 0, floor( fdiv( real( n ), 3 ) ) + 1 ):
result = fadd( result, fprod( [ power( -1, j ), binomial( fadd( n, 1 ), j ),
binomial( fsub( fmul( 2, n ), fmul( 3, j ) ), n ) ] ) )
return fdiv( result, fadd( n, 1 ) )
def getNthAperyNumber( n ):
result = 0
for k in arange( 0, real( n ) + 1 ):
result = fadd( result, fmul( power( binomial( n, k ), 2 ),
power( binomial( fadd( n, k ), k ), 2 ) ) )
return result
def exponentiatePolynomialOperator( n, k ):
'''Exponentiates an arbitrary polynomial by an integral power k.'''
result = n
for _ in arange( 0, k - 1 ):
result = multiplyPolynomials( result, n )
return result
def getThueMorseConstant( ):
result = 0
factor = mpmathify( '0.5' )
for i in arange( 0, mp.prec + 1 ):
result = fadd( result, fmul( getNthThueMorse( i ), factor ) )
factor = fdiv( factor, 2 )
return result
def getNthPolytopeNumber( n, d ):
result = real_int( n )
m = n + 1
for i in arange( 1, d - 1 ):
result = fmul( result, m )
m += 1
return fdiv( result, fac( d - 1 ) )
def rollMultipleDice( expression, times ):
result = [ ]
dice = parseDiceExpression( expression )
for i in arange( 0, times ):
result.append( evaluateDiceExpression( dice ) )
return result
def getThueMorseConstant( ):
result = 0
factor = mpmathify( '0.5' )
for i in arange( 0, mp.prec + 1 ):
result = fadd( result, fmul( getNthThueMorseNumber( i ), factor ) )
factor = fdiv( factor, 2 )
return result
def getNthPolytopeNumberOperator(n, d):
result = n
m = fadd(n, 1)
for _ in arange(1, d):
result = fmul(result, m)
m = fadd(m, 1)
return fdiv(result, fac(d))
def getExpandedFactorListSympy( factors ):
'''Takes a list of tuples where each tuple is a prime factor and an exponent
and returns a simple list of prime factors.'''
result = [ ]
for key in factors:
for i in arange( 0, factors[ key ] ):
result.append( key )
return sorted( result )
def getNthMerten( n ):
if real( n ) == 1:
return 1
result = 0
for i in arange( 1, n + 1 ):
result = fadd( result, getMobius( i ) )
return result
def PlotPos(self, fignum=None, color='g', dt=0.01, lw=2):
tVect = arange(0, self.duration, dt)
if tVect[-1] < self.duration:
tVect = np.append(tVect, self.duration)
xVect = [self.EvalPos(t) for t in tVect]
if fignum is not None:
plt.figure(fignum)
plt.plot(tVect, xVect, color=color, linewidth=lw)
plt.show(False)
def isBaseKPandigitalOperator(n, base):
digits = convertToBaseN(n, base, outputBaseDigits=True)
for i in arange(min(int(base), len(digits))):
try:
digits.index(i)
except ValueError:
return 0
return 1
def getNthPrimorial( n ):
if real_int( n ) == 0:
return 1
result = 2
for i in arange( 1, n ):
result = fmul( result, getNthPrime( i + 1 ) )
return result
def getNthReversalAdditionGenerator(n, k):
nextOne = int(n)
yield nextOne
for _ in arange(k):
nextOne = fadd(reverseDigits(nextOne), nextOne)
yield nextOne
if isPalindrome(nextOne):
break
def getPreviousPrimes( p, k, func = getPreviousPrimeCandidateForAny ):
if p < 1000000000 and getNthPrime( p ) < k:
raiseError( 'There is no previous prime to 2.' )
result = [ ]
for i in arange( 0, k ):
p = getPreviousPrime( p, func )
result.append( p )
return result
def getNthReversalAddition( n, k ):
next = int( real_int( n ) )
yield next
previous = next
for i in arange( k ):
if isPalindrome( next ):
break
next = fadd( reverseDigits( next ), next )
yield next
def getNthSylvester( n ):
if real( n ) == 1:
return 2
elif n == 2:
return 3
else:
list = [ 2, 3 ]
for i in arange( 2, n ):
list.append( fprod( list ) + 1 )
return list[ -1 ]
def getNthMotzkinNumber( n ):
'''
http://oeis.org/A001006
a(n) = sum((-1)^j*binomial(n+1, j)*binomial(2n-3j, n), j=0..floor(n/3))/(n+1)
'''
result = 0
for j in arange( 0, floor( fdiv( real( n ), 3 ) ) + 1 ):
result = fadd( result, fprod( [ power( -1, j ), binomial( fadd( n, 1 ), j ),
binomial( fsub( fmul( 2, n ), fmul( 3, j ) ), n ) ] ) )
return fdiv( result, fadd( n, 1 ) )
def getNthAperyNumber( n ):
'''
http://oeis.org/A005259
a(n) = sum(k=0..n, C(n,k)^2 * C(n+k,k)^2 )
'''
result = 0
for k in arange( 0, real( n ) + 1 ):
result = fadd( result, fmul( power( binomial( n, k ), 2 ),
power( binomial( fadd( n, k ), k ), 2 ) ) )
return result
def tetrateRightOperator(i, j):
'''
This is the larger, right-associative version of the hyper4 operator.
This function forces the second argument to an integer and runs in O( n )
time based on the second argument.
'''
result = i
for _ in arange(1, j):
result = power(i, result)
return result
def tetrateOperator(i, j):
'''
This is the smaller (left-associative) version of the hyper4 operator.
This function forces the second argument to an integer and runs in O( n )
time based on the second argument.
'''
result = i
for _ in arange(1, j):
result = power(result, i)
return result
def exponentiatePolynomial( n, k ):
if not isinstance( n, list ):
n = [ n ]
if isinstance( k, list ):
return [ multiplyPolynomials( n, i ) for i in k ]
else:
result = n
for i in arange( 0, k - 1 ):
result = multiplyPolynomials( result, n )
return result
def getNthSylvesterNumber(n):
if n == 1:
return 2
if n == 2:
return 3
sylvesters = [2, 3]
for _ in arange(2, n):
sylvesters.append(fprod(sylvesters) + 1)
return sylvesters[-1]
def getNthAlternatingFactorial( n ):
result = 0
negative = False
for i in arange( real( n ), 0, -1 ):
if negative:
result = fadd( result, fneg( fac( i ) ) )
negative = False
else:
result = fadd( result, fac( i ) )
negative = True
return result
def exponentiatePolynomial( n, k ):
'''Exponentiates an arbitrary polynomial by an integral power k.'''
if not isinstance( n, list ):
n = [ n ]
if isinstance( k, list ):
return [ multiplyPolynomials( n, i ) for i in k ]
else:
result = n
for i in arange( 0, k - 1 ):
result = multiplyPolynomials( result, n )
return result
def getAliquotSequence( n, k ):
'''
The aliquot sum of n is the sum of the divisors of n, not counting n itself
as a divisor. Subsequent aliquot sums can then be computed. These sequences
usually terminate, but some, like 276, get so large it has not been determined
if they ever terminate.
'''
yield real( floor( n ) )
a = n
for i in arange( 0, real( k ) - 1 ):
b = fsub( getSigma( a ), a )
yield b
a = b
def getNthDelannoyNumber(n):
if n == 1:
return 3
precision = int(fmul(n, 0.8))
if mp.dps < precision:
mp.dps = precision
result = 0
for k in arange(0, fadd(n, 1)):
result = fadd(result, fmul(binomial(n, k), binomial(fadd(n, k), k)))
return result
def getNthSchroederNumber( n ):
if real( n ) == 1:
return 1
if n < 0:
raise ValueError( '\'nth_schroeder\' expects a non-negative argument' )
n = fsub( n, 1 )
result = 0
for k in arange( 0, fadd( n, 1 ) ):
result = fadd( result, fdiv( fprod( [ power( 2, k ), binomial( n, k ),
binomial( n, fsub( k, 1 ) ) ] ), n ) )
return result
def PlotPos(self, fignum=None, color='g', dt=0.01, lw=2, includingSW=False):
tVect = arange(0, self.duration, dt)
if tVect[-1] < self.duration:
tVect = np.append(tVect, self.duration)
xVect = [self.EvalPos(t) for t in tVect]
if fignum is not None:
plt.figure(fignum)
plt.plot(tVect, xVect, color=color, linewidth=lw)
if includingSW:
ax = plt.gca().axis()
for s in self.switchpointsList:
plt.plot([s, s], [ax[2], ax[3]], 'r', linewidth=1)
plt.show(False)
def PlotPos(self, fignum='Displacement Profiles', includingSW=False, dt=0.005):
plt.figure(fignum)
tVect = arange(0, self.duration, dt)
if tVect[-1] < self.duration:
tVect = np.append(tVect, self.duration)
xVect = [self.EvalPos(t) for t in tVect]
plt.plot(tVect, xVect, linewidth=2)
handle = ['joint {0}'.format(i + 1) for i in xrange(self.ndof)]
plt.legend(handle)
if includingSW:
ax = plt.gca().axis()
for s in self.switchpointsList:
plt.plot([s, s], [ax[2], ax[3]], 'r', linewidth=1)
plt.show(False)
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 getNthAperyNumber(n):
'''
http://oeis.org/A005259
a(n) = sum(k=0..n, C(n,k)^2 * C(n+k,k)^2 )
'''
precision = int(fmul(n, 1.6))
if mp.dps < precision:
mp.dps = precision
result = 0
for k in arange(0, n + 1):
result = fadd(
result,
fmul(power(binomial(n, k), 2), power(binomial(fadd(n, k), k), 2)))
return result
def getNthSchroederHipparchusNumber(n):
'''
https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number
https://oeis.org/A001003
'''
if n == 0:
return 1
precision = int(fmul(n, 0.8))
if mp.dps < precision:
mp.dps = precision
result = 0
for i in arange(n):
result = fadd(result,
fmul(getNarayanaNumber(n, fadd(i, 1)), power(2, i)))
return result
def isKthPower(n, k):
if not isint(k, gaussian=True):
raise ValueError('integer argument expected')
if k == 1:
return 1
if im(n):
# I'm not sure why this is necessary...
if re(n) == 0:
return isKthPower(im(n), k)
# We're looking for a Gaussian integer among any of the roots.
for i in [autoprec(root)(n, k, i) for i in arange(k)]:
if isint(i, gaussian=True):
return 1
return 0
rootN = autoprec(root)(n, k)
return 1 if isint(rootN, gaussian=True) else 0
def getNthSchroederNumber(n):
if n == 1:
return 1
n = fsub(n, 1)
result = 0
precision = int(fmul(n, 0.8))
if mp.dps < precision:
mp.dps = precision
for k in arange(0, fadd(n, 1)):
result = fadd(
result,
fdiv(fprod([power(2, k),
binomial(n, k),
binomial(n, fsub(k, 1))]), n))
return result
def getNthMotzkinNumber(n):
'''
http://oeis.org/A001006
a(n) = sum((-1)^j*binomial(n+1, j)*binomial(2n-3j, n), j=0..floor(n/3))/(n+1)
'''
precision = int(n)
if mp.dps < precision:
mp.dps = precision
result = 0
for j in arange(0, floor(fdiv(n, 3)) + 1):
result = fadd(
result,
fprod([
power(-1, j),
binomial(fadd(n, 1), j),
binomial(fsub(fmul(2, n), fmul(3, j)), n)
]))
return fdiv(result, fadd(n, 1))
"""
Module: AWD, Analysis, wissenschaftliches Rechnen und Datenvisualisierung
Course: BSc INF 2015, ZH5-Mo, FS16, Schuler Josef A.
This is my solution for the eight of eight parts for the module at FFHS. This shows
some examples of how to visualize the "Kepler'sche Fassregel" and the "Trapezregel"
"""
from mpmath import arange
from sympy import *
x = Symbol('x')
domain = arange(-12, 12, 0.1)
function = x**3
def do_trapezregel(fn, N, a, b):
"""
Calculate integral using the 'Trapezregel'. This function slices the domain in
N equal portions and calculates their integrals. It also takes negative integrals
into account and adds the corresponding zero-positions to the sections.
:param fn: Function to integrate
:param N: Number of slices (resolution)
:param a: Left bound
:param b: Right bound
:return: Integral
"""
title = 'Trapezregel:'
print('\n\n{title}\n{underline}'.format(title=title,
underline=len(title) * '='))
# Find 0-positions
zero_positions = solve(fn, x)
def getNthMultifactorialOperator(n, k):
return fprod(arange(n, 0, -(k)))
def getSublistGenerator( args, start, count ):
for i in arange( start, fadd( start, count ) ):
yield args[ int( i ) ]
def describeIntegerOperator(n):
indent = ' ' * 4
print()
print(int(n), 'is:')
if isOdd(n):
print(indent + 'odd')
elif isEven(n):
print(indent + 'even')
if isPrime(n):
isPrimeFlag = True
print(indent + 'prime')
elif n > 3:
isPrimeFlag = False
print(indent + 'composite')
else:
isPrimeFlag = 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),
minimum=1)
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),
maximum=50)
if result[0]:
print(indent + 'the ' + getShortOrdinalName(result[1]) +
' Mersenne exponent')
if not isPrimeFlag and n != 1 and n <= LARGEST_NUMBER_TO_FACTOR:
# 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')
# 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 isKaprekarNumber(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, log)
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 isPrimeFlag and n != 1 and n <= LARGEST_NUMBER_TO_FACTOR:
# 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 <= LARGEST_NUMBER_TO_FACTOR:
print(indent + 'a divisor sum of ' + str(int(getSigma(n))))
print(indent + 'a Stern value of ' + str(int(getNthSternNumber(n))))
calkinWilf = getNthCalkinWilf(n)
print(indent + 'a Calkin-Wilf value of ' + str(int(calkinWilf[0])) +
'/' + str(int(calkinWilf[1])))
print(indent + 'a Mobius value of ' + str(int(getNthMobiusNumber(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 getRandomIntegers(n, k):
'''
Returns k random integers between 0 and n-1.
'''
for _ in arange(0, k):
yield randrange(n)
def getMultipleRandoms(n):
'''
Returns n random numbers.
'''
for _ in arange(0, n):
yield rand()
def arange(*args):
"""Returns a numpy array with n repetitions of mp.mpf(0)"""
return np.array(mp.arange(*args))
#coding=utf-8 import math import pandas from mpmath import arange, plot print(math.factorial(4)) plot(arange(5))
def enumerateMultipleDice(expression, count):
dice = parseDiceExpression(expression)
for _ in arange(0, count):
yield evaluateDiceExpression(dice)[0]
