A354799 - OEIS

%I #37 Jun 29 2022 01:50:08

%S 16,81,128,144,324,400,432,625,648,784,1024,1152,1296,1936,2000,2025,

%T 2187,2401,2500,2592,2704,3200,3456,3600,3888,3969,4624,5000,5184,

%U 5488,5625,5776,6272,7056,8100,8192,8464,8748,9216,9604,9801,10000,10125,10368,10800

%N Integers m in A001694 such that 3 | d(m^2), where d(n) = A000005(n).

%H Michael S. Branicky, <a href="/A354799/b354799.txt">Table of n, a(n) for n = 1..10000</a>

%F Equals { A001694 \ A350014 }.

%F Equals { m in A001694 : d(m^2) mod 3 = 0 }.

%F Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - 5*zeta(3)/(2*zeta(2)) = 0.1166890133... . - _Amiram Eldar_, Jun 28 2022

%e A001694(5) = 16 is a term since d(16^2) = d(256) = 9, and 9 is a multiple of 3.

%e A001694(13) = 81 is a term since d(81^2) = d(6561) = 9, and 9 is a multiple of 3.

%e A001694(3) = 8 is not a term since d(8^2) = d(64) = 7, which is not divisible by 3.

%t With[{nn = 10800}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Mod[DivisorSigma[0, #^2], 3] == 0 &]]

%o (PARI) isok(m) = ispowerful(m) && !(numdiv(m^2) % 3); \\ _Michel Marcus_, Jun 27 2022

%o (Python)

%o from sympy import divisor_count as d, factorint as f

%o def ok(k): return k > 1 and min(f(k).values()) > 1 and d(k*k)%3 == 0

%o print([k for k in range(11000) if ok(k)]) # _Michael S. Branicky_, Jun 28 2022

%Y Cf. A000005, A001694, A008585, A076400, A350014.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Jun 21 2022