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

1,1

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

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)

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A366854 A380857 A217584 * A189988 A232892 A034285

Adjacent sequences: A030630 A030631 A030632 * A030634 A030635 A030636

Keyword

nonn,changed

Author

Jeff Burch

Status

approved