A350014 Numbers whose square has a number of divisors coprime to 6.

1, 4, 8, 9, 25, 27, 32, 36, 49, 64, 72, 100, 108, 121, 125, 169, 196, 200, 216, 225, 243, 256, 288, 289, 343, 361, 392, 441, 484, 500, 512, 529, 576, 675, 676, 729, 800, 841, 864, 900, 961, 968, 972, 1000, 1089, 1125, 1156, 1225, 1323, 1331, 1352, 1369, 1372, 1444

Offset: 1

Author

Michael De Vlieger, Jan 17 2022

Keywords

• nonn
• easy

Comments

a(n) = m in A001694 such that d(m^2) is not divisible by 3, where d(n) = A000005(n).

Supersequence of A051676 (composite numbers whose square has a prime number of divisors).

Subsequence of A001694 (powerful numbers).

Numbers whose prime factorization has only exponents that are congruent to {0, 2} mod 3 (A007494). - Amiram Eldar, Mar 31 2022

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000290, A007494, A001694, A001651, A051676, A240976.

Programs

Maple

A350014 := proc(n)
    option remember ;
    local a;
    if n =1 then
        1;
    else
        for a from procname(n-1)+1 do
            if igcd(numtheory[tau](a^2),6) = 1 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A350014(n),n=1..20) ; # R. J. Mathar, Apr 06 2022

Mathematica

Select[Range[1500], CoprimeQ[DivisorSigma[0, #^2], 6] &] (* or *)
With[{nn = 1500}, 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) = gcd(numdiv(m^2), 6) == 1; \\ Michel Marcus, Mar 04 2022

Formula

a(n) = {m : gcd(d(m^2), 6) = 1}.

Sum_{n>=1} 1/a(n) = 15*zeta(3)/Pi^2 (= 10 * A240976). - Amiram Eldar, Mar 31 2022