A175745 Numbers with 35 divisors.

5184, 11664, 40000, 153664, 250000, 455625, 937024, 1265625, 1750329, 1827904, 1882384, 5345344, 8340544, 9529569, 10673289, 17909824, 20820969, 28344976, 37515625, 45265984, 59105344, 60886809, 73530625, 77228944, 95004009, 119946304, 143496441, 180848704

Offset: 1

Author

Jaroslav Krizek, Aug 27 2010

Keywords

• nonn

Comments

Numbers of the form p^34 and p^6*q^4 (A190464), where p and q are distinct primes.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• OEIS Wiki, Index entries for number of divisors

Crossrefs

Cf. A000005, A175742, A175743, A175744.

Programs

Mathematica

Select[Range[9000000],DivisorSigma[0,#]==35&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

PARI

is(n)=numdiv(n)==35 \\ Charles R Greathouse IV, Jun 19 2016

Python

from sympy import primepi, integer_nthroot, primerange
def A175745(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(integer_nthroot(x//p**6,4)[0]) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,10)[0])-primepi(integer_nthroot(x,34)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

Formula

A000005(a(n)) = 35.

Sum_{n>=1} 1/a(n) = P(4)*P(6) - P(10) + P(34) = 0.000320676..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

Extensions

Extended by T. D. Noe, May 08 2011