A272190 Either 6th power of a prime, or product of the square of two different primes.

36, 64, 100, 196, 225, 441, 484, 676, 729, 1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 11236, 12321, 13225, 13924, 14161, 14884, 15129, 15625, 16641, 17689, 17956, 19881

Offset

1,1

Comments

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from Paolo P. Lava)

Formula

Sum_{n>=1} 1/a(n) = (P(2)^2 - P(4))/2 + P(6) = (A085548^2 - A085964)/2 + A085966 = 0.080837..., where P is the prime zeta function. - Amiram Eldar, Oct 03 2023

Example

36 = 2^2 * 3^2;  64 = 2^6.

Maple

with(numtheory): P:=proc(q) local a, k, n;  for n from 2 to q do a:=sort([op(divisors(n))]);
if 3*tau(n)= add(tau(a[k]), k=1..nops(a)-1) then print(n); fi; od; end: P(10^7);

Mathematica

Select[Range[20000], MemberQ[{{6}, {2, 2}}, FactorInteger[#][[;; , 2]]] &] (* Amiram Eldar, Oct 03 2023 *)

Prog

(PARI) isok(n) = 3*numdiv(n) == sumdiv(n, d, (n!=d)*numdiv(d)); \\ Michel Marcus, Apr 22 2016
(PARI) is(n) = {my(e = factor(n)[, 2]~); e == [6] || e == [2, 2]; } \\ Amiram Eldar, Oct 03 2023

Crossrefs

Cf. A030516, A080258, A085986, A272191.

Cf. A085548, A085964, A085966.

Sequence in context: A082295 A343293 A326666 * A060671 A269499 A074315

Adjacent sequences: A272187 A272188 A272189 * A272191 A272192 A272193

Keyword

nonn,easy

Author

Paolo P. Lava, Apr 22 2016

Status

approved