A272190 - OEIS

%I #16 Oct 03 2023 13:19:41

%S 36,64,100,196,225,441,484,676,729,1089,1156,1225,1444,1521,2116,2601,

%T 3025,3249,3364,3844,4225,4761,5476,5929,6724,7225,7396,7569,8281,

%U 8649,8836,9025,11236,12321,13225,13924,14161,14884,15129,15625,16641,17689,17956,19881

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

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

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

%p with(numtheory): P:=proc(q) local a,k,n;  for n from 2 to q do a:=sort([op(divisors(n))]);

%p if 3*tau(n)= add(tau(a[k]),k=1..nops(a)-1) then print(n); fi; od; end: P(10^7);

%t Select[Range[20000], MemberQ[{{6}, {2, 2}}, FactorInteger[#][[;; , 2]]] &] (* _Amiram Eldar_, Oct 03 2023 *)

%o (PARI) isok(n) = 3*numdiv(n) == sumdiv(n, d, (n!=d)*numdiv(d)); \\ _Michel Marcus_, Apr 22 2016

%o (PARI) is(n) = {my(e = factor(n)[, 2]~); e == [6] || e == [2, 2];} \\ _Amiram Eldar_, Oct 03 2023

%Y Cf. A030516, A080258, A085986, A272191.

%Y Cf. A085548, A085964, A085966.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, Apr 22 2016