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A350014
Numbers whose square has a number of divisors coprime to 6.
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
1,2
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
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
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 &]]
PROG
(PARI) isok(m) = gcd(numdiv(m^2), 6) == 1; \\ Michel Marcus, Mar 04 2022
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Jan 17 2022
STATUS
approved