A376172 Numbers whose prime factorization has an even minimum exponent.

1, 4, 9, 16, 25, 36, 49, 64, 72, 81, 100, 108, 121, 144, 169, 196, 200, 225, 256, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 625, 675, 676, 729, 784, 800, 841, 900, 961, 968, 972, 1024, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1369, 1372, 1444

Offset: 1

Author

Amiram Eldar, Sep 13 2024

Keywords

• nonn
• easy

Comments

Numbers k such that A051904(k) is even.

The minimum exponent in the prime factorization of 1 is considered to be A051904(1) = 0, and therefore 1 is a term of this sequence.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to powerful numbers.

Crossrefs

Subsequence of A001694.

Complement of A376173 within A001694.

Subsequences: A001248, A062503, A325240.

Cf. A051904.

Programs

Mathematica

seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # == 1 || EvenQ[Min[FactorInteger[#][[;; , 2]]]] &]; seq[2000]

PARI

is(k) = {my(f = factor(k), e = f[,2]); !(#e) || (ispowerful(f) && !(vecmin(e) % 2));}

Formula

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} (-1)^(k+1) * s(k) = 1.70559662202357112914..., where s(k) = Product_{p prime} (1 + 1/(p^k*(p-1))).