A137490 Numbers with 27 divisors.

900, 1764, 2304, 4356, 4900, 6084, 6400, 10404, 11025, 12100, 12544, 12996, 16900, 19044, 23716, 26244, 27225, 28900, 30276, 30976, 33124, 34596, 36100, 38025, 43264, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 73984, 74529

Offset

1,1

Comments

Maple implementation: see A030513.

Numbers of the form p^26 (subset of A089081), p^2*q^2*r^2 (like 900, 1764, 4356, squares of A007304) or p^2*q^8 (like 2304, 6400, subset of the squares of A030628) where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

A000005(a(n)) = 27.

Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 + P(2)*P(8) - P(10) + P(26) = 0.00453941..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

Mathematica

Select[Range[150000], DivisorSigma[0, #]==27&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

Prog

(PARI) isA137490(n) = numdiv(n)==27 \\ Michael B. Porter, Apr 10 2010
(Python)
from math import isqrt
from sympy import integer_nthroot, primerange, primepi
from oeis_sequences.OEISsequences import bisection
def A137490(n):
    def f(x):
        c = n+x
        c += -sum(primepi(isqrt(x//(k*m)**2))-b for a, k in enumerate(primerange(integer_nthroot(x, 6)[0]+1), 1) for b, m in enumerate(primerange(k+1, integer_nthroot(x//(k*k), 4)[0]+1), a+1))
        c += -sum(primepi(isqrt(x//p**8)) for p in primerange(integer_nthroot(x, 8)[0]+1))+primepi(integer_nthroot(x, 10)[0])
        c += -primepi(integer_nthroot(x, 26)[0])
        return int(c)
    return bisection(f, n, n) # Chai Wah Wu, May 01 2026

Crossrefs

Cf. A000005, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Sequence in context: A318720 A338540 A390950 * A129575 A328136 A336254

Adjacent sequences: A137487 A137488 A137489 * A137491 A137492 A137493

Keyword

nonn

Author

R. J. Mathar, Apr 22 2008

Status

approved