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

1,1

Comments

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

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A030516, A143610, A179645, A272190.

Cf. A085548, A085541, A085965, A085968.

Sequence in context: A356871 A359280 A307758 * A072412 A052486 A114128

Adjacent sequences: A272188 A272189 A272190 * A272192 A272193 A272194

Keyword

nonn,easy

Author

Paolo P. Lava, Apr 22 2016

Status

approved