cp's OEIS Frontend

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

%I A272190 #16 Oct 03 2023 13:19:41
%S A272190 36,64,100,196,225,441,484,676,729,1089,1156,1225,1444,1521,2116,2601,
%T A272190 3025,3249,3364,3844,4225,4761,5476,5929,6724,7225,7396,7569,8281,
%U A272190 8649,8836,9025,11236,12321,13225,13924,14161,14884,15129,15625,16641,17689,17956,19881
%N A272190 Either 6th power of a prime, or product of the square of two different primes.
%C A272190 Numbers such that the sum of the number of divisors of their aliquot parts is three times the number of their divisors.
%H A272190 Amiram Eldar, <a href="/A272190/b272190.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..200 from Paolo P. Lava)
%F A272190 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
%e A272190 36 = 2^2 * 3^2;  64 = 2^6.
%p A272190 with(numtheory): P:=proc(q) local a,k,n;  for n from 2 to q do a:=sort([op(divisors(n))]);
%p A272190 if 3*tau(n)= add(tau(a[k]),k=1..nops(a)-1) then print(n); fi; od; end: P(10^7);
%t A272190 Select[Range[20000], MemberQ[{{6}, {2, 2}}, FactorInteger[#][[;; , 2]]] &] (* _Amiram Eldar_, Oct 03 2023 *)
%o A272190 (PARI) isok(n) = 3*numdiv(n) == sumdiv(n, d, (n!=d)*numdiv(d)); \\ _Michel Marcus_, Apr 22 2016
%o A272190 (PARI) is(n) = {my(e = factor(n)[, 2]~); e == [6] || e == [2, 2];} \\ _Amiram Eldar_, Oct 03 2023
%Y A272190 Cf. A030516, A080258, A085986, A272191.
%Y A272190 Cf. A085548, A085964, A085966.
%K A272190 nonn,easy
%O A272190 1,1
%A A272190 _Paolo P. Lava_, Apr 22 2016