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

%I A272191 #13 Oct 03 2023 13:14:00
%S A272191 72,108,200,256,392,500,675,968,1125,1323,1352,1372,2312,2888,3087,
%T A272191 3267,4232,4563,5324,6125,6561,6728,7688,7803,8575,8788,9747,10952,
%U A272191 11979,13448,14283,14792,15125,17672,19652,19773,21125,22472,22707,25947,27436,27848,29768
%N A272191 Either 8th power of a prime, or product of a square and a cube of two different primes.
%C A272191 Numbers such that the sum of the number of divisors of their aliquot parts is four times the number of their divisors.
%H A272191 Amiram Eldar, <a href="/A272191/b272191.txt">Table of n, a(n) for n = 1..10000</a>
%F A272191 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
%e A272191 72 = 2^3 * 3^2;  256 = 2^8.
%p A272191 with(numtheory): P:=proc(q) local a,k,n;  for n from 2 to q do a:=sort([op(divisors(n))]);
%p A272191 if 4*tau(n)= add(tau(a[k]),k=1..nops(a)-1) then print(n); fi; od; end: P(10^7);
%t A272191 Select[Range[30000], MemberQ[{{8}, {2, 3}}, Sort[FactorInteger[#][[;; , 2]]]] &] (* _Amiram Eldar_, Oct 03 2023 *)
%o A272191 (PARI) isok(n) = 4*numdiv(n) == sumdiv(n, d, (n!=d)*numdiv(d)); \\ _Michel Marcus_, Apr 22 2016
%o A272191 (PARI) is(n) = {my(e = vecsort(factor(n)[, 2])~); e == [8] || e == [2, 3];} \\ _Amiram Eldar_, Oct 03 2023
%Y A272191 Cf. A030516, A143610, A179645, A272190.
%Y A272191 Cf. A085548, A085541, A085965, A085968.
%K A272191 nonn,easy
%O A272191 1,1
%A A272191 _Paolo P. Lava_, Apr 22 2016