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from math import isqrt, prod
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(PARI) k=14; start=2^k; finish=240000; v=[] ; for(n=start, finish, if(bigomega(n)==k, v=concat(v, n))); v
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return bisection(f, n, n) # Chai Wah Wu, Nov 03 2024
(Python)
from math import isqrt
from sympy import primerange, integer_nthroot, primepi
def A069275(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b, isqrt(x//c)+1), a)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b, integer_nthroot(x//c, m)[0]+1), a) for d in g(x, a2, b2, c*b2, m-1)))
def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x, 0, 1, 1, 14)))
return bisection(f, n, n) # Chai Wah Wu, Nov 03 2024
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