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

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 three times the number of their divisors.

Examples

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

Links

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

Crossrefs

Cf. A030516, A080258, A085986, A272191.

Cf. A085548, A085964, A085966.

Programs

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 *)

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

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