A139118 Numbers with a nonprime number of divisors.

1, 6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102

Offset: 1

Author

Omar E. Pol, May 09 2008

Keywords

• easy
• nonn

Comments

A000005(a(n)) is nonprime. Complement of A009087. Also, nonprime numbers with nonprime number of divisors.

The sequence consists of those n such that n is not a prime power, or n of the form p^k where k+1 is composite. - Franklin T. Adams-Watters, Apr 09 2009

Examples

			15 is in the sequence because it has 4 divisors: 1, 3, 5, and 15. - _Emeric Deutsch_, Jun 27 2009
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A009087, A018252.

Programs

Maple

with(numtheory): a := proc (n) if isprime(tau(n)) = false then n else end if end proc: seq(a(n), n = 1 .. 120); # Emeric Deutsch, Jun 27 2009

Mathematica

Select[Range[102], ! PrimeQ[DivisorSigma[0, #]] &] (* Amiram Eldar, Nov 27 2020 *)

PARI

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

Python

from sympy import primepi, integer_nthroot, primerange
def A139118(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k-1)[0]) for k in primerange(x.bit_length()+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 22 2025

Extensions

Extended by Ray Chandler, Jun 25 2009