A030633 Numbers with 15 divisors.

144, 324, 400, 784, 1936, 2025, 2500, 2704, 3969, 4624, 5625, 5776, 8464, 9604, 9801, 13456, 13689, 15376, 16384, 21609, 21904, 23409, 26896, 29241, 29584, 30625, 35344, 42849, 44944, 55696, 58564, 59536, 60025, 68121, 71824, 75625

Offset: 1

Author

Jeff Burch

Keywords

• nonn

Comments

Numbers of the form p^14 (subset of A010802) or p^2*q^4 (A189988) where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A010802, A030630, A030631, A030632, A189988.

Programs

Mathematica

Select[Range[300000],DivisorSigma[0,#]==15&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

PARI

is(n)=numdiv(n)==15 \\ Charles R Greathouse IV, Jun 19 2016

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A030633(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(isqrt(x//p**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,6)[0])-primepi(integer_nthroot(x,14)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

Formula

From Amiram Eldar, Jul 03 2022: (Start)

A000005(a(n)) = 15.

Sum_{n>=1} 1/a(n) = P(2)*P(4) - P(6) + P(14) = 0.0178111..., where P is the prime zeta function. (End)