A175744 Numbers with 34 divisors.

196608, 327680, 458752, 720896, 851968, 1114112, 1245184, 1507328, 1900544, 2031616, 2424832, 2686976, 2818048, 3080192, 3473408, 3866624, 3997696, 4390912, 4653056, 4784128, 5177344, 5439488, 5832704, 6356992, 6619136, 6750208, 7012352, 7143424, 7405568

Offset: 1

Author

Jaroslav Krizek, Aug 27 2010

Keywords

• nonn

Comments

Numbers of the form p^33 and p^16*q^1, 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. A175742, A175743.

Programs

Mathematica

Select[Range[6000000],DivisorSigma[0,#]==34&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

PARI

is(n)=numdiv(n)==34 \\ Charles R Greathouse IV, Jun 19 2016

Python

from sympy import primepi, primerange, integer_nthroot
def A175744(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 int(n+x-sum(primepi(x//p**16) for p in primerange(integer_nthroot(x,16)[0]+1))+primepi(integer_nthroot(x,17)[0])-primepi(integer_nthroot(x,33)[0]))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

Formula

A000005(a(n))=34.