A030634 Numbers with 16 divisors.

120, 168, 210, 216, 264, 270, 280, 312, 330, 378, 384, 390, 408, 440, 456, 462, 510, 520, 546, 552, 570, 594, 616, 640, 680, 690, 696, 702, 714, 728, 744, 750, 760, 770, 798, 858, 870, 888, 896, 910, 918, 920, 930, 945, 952, 966, 984, 1000

Offset

1,1

Comments

Numbers of the form p^15 (subset of A010803), p*q^7, p*q*r^3 or p^3*q^3, or p*q*r*s, where p, q, r and s are distinct primes. - R. J. Mathar, Mar 01 2010

Links

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

Jérôme Germoni, Nombres à huit diviseurs, Images des Mathématiques, CNRS, 2017 (in French).

Mathematica

Select[Range[3000], DivisorSigma[0, #]==16&] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)

Prog

(PARI) is(n)=numdiv(n)==16 \\ Charles R Greathouse IV, Jun 19 2016
(Magma) [n: n in [1..1000] | DivisorSigma(0, n) eq 16]; // Vincenzo Librandi, Oct 05 2017
(Python)
from math import isqrt
from sympy import integer_nthroot, primerange, primepi
from oeis_sequences.OEISsequences import bisection
def A030634(n):
    def f(x):
        x5 = integer_nthroot(x, 5)[0]+1
        c = n+x
        c += -sum(primepi(integer_nthroot(x//k**3, 3)[0])-a for a, k in enumerate(primerange(integer_nthroot(x, 6)[0]+1), 1))
        c += -sum(primepi(x//(k*m*r))-c for a, k in enumerate(primerange(integer_nthroot(x, 4)[0]+1), 1) for b, m in enumerate(primerange(k+1, isqrt(x//k)+1), a+1) for c, r in enumerate(primerange(m+1, isqrt(x//(k*m))+1), b+1))
        c += -sum(primepi(x//(k**3*m))-b for a, k in enumerate(primerange(x5), 1) for b, m in enumerate(primerange(k+1, isqrt(x//(k**3))+1), a+1))
        c += -sum(primepi(x//(k*m**3))-b for a, k in enumerate(primerange(x5), 1) for b, m in enumerate(primerange(k+1, integer_nthroot(x//k, 4)[0]+1), a+1))
        c += -sum(primepi(integer_nthroot(x//(k*m), 3)[0])-b for a, k in enumerate(primerange(x5), 1) for b, m in enumerate(primerange(k+1, integer_nthroot(x//k, 4)[0]+1), a+1))
        c += -sum(primepi(x//p**7) for p in primerange(integer_nthroot(x, 7)[0]+1))+primepi(integer_nthroot(x, 8)[0])
        c += -primepi(integer_nthroot(x, 15)[0])
        return int(c)
    return bisection(f, n, n) # Chai Wah Wu, May 01 2026

Crossrefs

Cf. A030626, A030631, A030632, A030633.

Sequence in context: A179642 A099832 A111399 * A396323 A272594 A377156

Adjacent sequences: A030631 A030632 A030633 * A030635 A030636 A030637

Keyword

nonn

Author

Jeff Burch

Status

approved