def pi_2(n: int) -> int:
ns = [
primepi(n / prime(i)) - i + 1 for i in range(1,
primepi(sqrt(n)) + 1)
]
return sum(ns)
def DpList(n):
D = []
for p in sy.primerange(1, n + 3):
m = 1
while (sy.primepi(p - m) == sy.primepi(p + m) - 1):
m = m + 1
D.append(m - 1)
return D
def opt_factorial_factors(factorial_n):
prime_factors = {}
for i in xrange(1, sympy.primepi(factorial_n) + 1):
p = sympy.prime(i)
k = prime_exponent(factorial_n, p)
prime_factors[p] = k
return prime_factors
def main(N=10**8):
c = 0
u = int(N**0.5)
primes = primerange(1, u + 1)
for n, p in enumerate(primes):
c += primepi((N - 1) // p) - n
return c
def main(N=5000):
i = 11
while True:
w = ways(i, primepi(i))
if w > N:
return i
else:
i += 1
def primepi(n):
"""Number of primes less than a value
http://mathworld.wolfram.com/PrimeCountingFunction.html
This is here as a reminder of what is in sympy
>>> primepi(10**5)
9592
"""
return sympy.primepi(n)
def primepi(n):
"""Number of primes less than a value
http://mathworld.wolfram.com/PrimeCountingFunction.html
This is here as a reminder of what is in sympy
>>> primepi(10**5)
9592
"""
return sympy.primepi(n)
def pi(n):
n = np.floor(n).astype(int)
if isinstance(n,(list,np.ndarray)):
out = []
for cur_num in n:
out.append(sp.primepi(cur_num))
return out
else:
out=0
for x in range(1,n):
if sp.isprime(x):
out+=1
return out
def method3_non_recursive(n):
primes = primesieve.primes(n)
SS = [[1]]
S = [1]
for k in range(1, n + 1):
if k % 5_000 == 0:
print(f"progress: {k}")
SSk = []
pi = sympy.primepi(k)
for i in range(pi):
p = primes[i]
SSkmp = SS[k - p]
SSk.append(((SSk[i - 1] if i > 0 else 0) + p *
(SSkmp[i] if i < len(SSkmp) else
(SSkmp[-1] if SSkmp else 0))) % MODULUS)
SS.append(SSk)
S.append(SSk[pi - 1] if pi >= 1 else 0)
def method3(n):
primes = primesieve.primes(n)
# S[k, i] is the sum of numbers whose prime factor sum is k and
# whose prime factors are among the primes[0], primes[1], ...,
# primes[i].
@functools.lru_cache(maxsize=None)
def S(k, i):
if k == 0:
return 1
if i < 0:
return 0
p = primes[i]
if k < p:
return S(k, i - 1)
else:
return (S(k, i - 1) + p * S(k - p, i)) % MODULUS
return [S(k, sympy.primepi(k) - 1) for k in range(n + 1)]
#!/usr/bin/python3
# -*- coding: utf-8 -*-
# A composite is a number containing at least two prime factors. For example, 15 = 3 × 5; 9 = 3 × 3; 12 = 2 × 2 × 3.
# There are ten composites below thirty containing precisely two, not necessarily distinct, prime factors: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26.
# How many composite integers, n < 10^8, have precisely two, not necessarily distinct, prime factors?
# 10^8'den yarısından küçük her asal için kendinden küçük eşit asal sayılarla çarpımından oluşturulan sayılar sayıldığında aranan sonuç bulunmuş oluyor.
from sympy import primepi
from sympy import primerange
sonuc = 0
LIMIT = 10**8
primes = primerange(2, LIMIT // 2)
for n in primes:
if (n < LIMIT**(0.5)):
m = primepi(n)
else:
m = primepi(LIMIT // n)
sonuc += m
print(sonuc)
def pi(x):
values = np.zeros(len(x))
for index, val in enumerate(x):
pi = sp.primepi(val)
values[index] = pi
return values
def division(x, func):
return np.divide(func(x), f(x))
# SECCIÓN PRINCIPAL
# ------------------------------------------------
x = int(sys.argv[1])
counter = 59
closure = True
while closure and counter <= x:
pi = sp.primepi(counter)
closure = pi > g(counter) and pi < h(counter)
counter += 1
if closure:
print("La función pi(x) con x <= " + str(x) + " se encuentra acotada.")
else:
print("La función pi(x) con x <= " + str(x) +
" no se encuentra acotada para el valor " + str(counter - 1))
import math import sympy x = 200 pi_x = sympy.primepi(x) print(pi_x)
# http://mathworld.wolfram.com/Semiprime.html
from __future__ import division
from sympy import prime, primepi
n = 10 ** 8
s = 0
for k in range(1, primepi(sqrt(n)) + 1):
s += primepi(n / prime(k)) - k + 1
print s
def pi(x):
values = np.zeros(len(x))
for index, val in enumerate(x):
pi = sp.primepi(val)
values[index] = pi
return values
def numSemiprimes(x):
base = list(sieve.primerange(1, int(x**0.5) + 1))
return sum([
primepi(int(x / base[k - 1]) + 1) - k + 1
for k in range(1, primepi(int(x**0.5)))
])
from sympy import primerange, primepi
import time
start = time.time()
n = 10**8
count = 0
for i in primerange(2, 10**4):
count += primepi(n / i) - primepi(i) + 1
if time.time() - start > 10:
print(i, count)
start = time.time()
print(count)
nextx = xcenter - width / children - dx / children
for neighbor in neighbors:
nextx += dx
pos = hierarchy_pos(G,
neighbor,
width=dx,
vert_gap=vert_gap,
vert_loc=vert_loc - vert_gap,
xcenter=nextx,
pos=pos,
parent=root)
return pos
K = 5
CHILDREN = 2**(primepi(K - 1))
M = 86
FONT_SIZE = 0
NODE_SIZE = 2
DPI = 300
ROOT = 0
list_ = []
chosen_numbers = [
5, 7, 9, 10, 11, 13, 14, 16, 17, 20, 21, 22, 23, 25, 26, 27, 30, 32, 34,
35, 36, 37, 39, 40, 41, 42, 44, 47, 49, 51, 52, 54, 56, 57, 58, 59, 61, 63,
64, 65, 66, 68, 71, 73, 76, 77, 79, 81, 83, 84
]
for M in chosen_numbers:
graph = general_collatz_tree(M, K, root=ROOT + 1)
