A354799 Integers m in A001694 such that 3 | d(m^2), where d(n) = A000005(n).

16, 81, 128, 144, 324, 400, 432, 625, 648, 784, 1024, 1152, 1296, 1936, 2000, 2025, 2187, 2401, 2500, 2592, 2704, 3200, 3456, 3600, 3888, 3969, 4624, 5000, 5184, 5488, 5625, 5776, 6272, 7056, 8100, 8192, 8464, 8748, 9216, 9604, 9801, 10000, 10125, 10368, 10800

Offset: 1

Author

Michael De Vlieger, Jun 21 2022

Keywords

• nonn
• easy

Examples

			A001694(5) = 16 is a term since d(16^2) = d(256) = 9, and 9 is a multiple of 3.
A001694(13) = 81 is a term since d(81^2) = d(6561) = 9, and 9 is a multiple of 3.
A001694(3) = 8 is not a term since d(8^2) = d(64) = 7, which is not divisible by 3.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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 &]]

PARI

isok(m) = ispowerful(m) && !(numdiv(m^2) % 3); \\ Michel Marcus, Jun 27 2022

Python

from sympy import divisor_count as d, factorint as f
def ok(k): return k > 1 and min(f(k).values()) > 1 and d(k*k)%3 == 0
print([k for k in range(11000) if ok(k)]) # Michael S. Branicky, Jun 28 2022

Formula

Equals { A001694 \ A350014 }.

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

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