A272191 Either 8th power of a prime, or product of a square and a cube of two different primes.

72, 108, 200, 256, 392, 500, 675, 968, 1125, 1323, 1352, 1372, 2312, 2888, 3087, 3267, 4232, 4563, 5324, 6125, 6561, 6728, 7688, 7803, 8575, 8788, 9747, 10952, 11979, 13448, 14283, 14792, 15125, 17672, 19652, 19773, 21125, 22472, 22707, 25947, 27436, 27848, 29768

Offset: 1

Author

Paolo P. Lava, Apr 22 2016

Keywords

• nonn
• easy

Comments

Numbers such that the sum of the number of divisors of their aliquot parts is four times the number of their divisors.

Examples

			72 = 2^3 * 3^2;  256 = 2^8.
		

Links

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

Crossrefs

Cf. A030516, A143610, A179645, A272190.

Cf. A085548, A085541, A085965, A085968.

Programs

Maple

with(numtheory): P:=proc(q) local a,k,n;  for n from 2 to q do a:=sort([op(divisors(n))]);
if 4*tau(n)= add(tau(a[k]),k=1..nops(a)-1) then print(n); fi; od; end: P(10^7);

Mathematica

Select[Range[30000], MemberQ[{{8}, {2, 3}}, Sort[FactorInteger[#][[;; , 2]]]] &] (* Amiram Eldar, Oct 03 2023 *)

PARI

isok(n) = 4*numdiv(n) == sumdiv(n, d, (n!=d)*numdiv(d)); \\ Michel Marcus, Apr 22 2016

PARI

is(n) = {my(e = vecsort(factor(n)[, 2])~); e == [8] || e == [2, 3];} \\ Amiram Eldar, Oct 03 2023

Formula

Sum_{n>=1} 1/a(n) = P(2)*P(3) - P(5) + P(8) = A085548 * A085541 - A085965 + A085968 = 0.047342..., where P is the prime zeta function. - Amiram Eldar, Oct 03 2023