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

1,2

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.

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))).

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]

Prog

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

Crossrefs

Subsequence of A001694.

Complement of A376173 within A001694.

Subsequences: A001248, A062503, A325240.

Cf. A051904.

Sequence in context: A292677 A292678 A072595 * A334832 A169669 A111707

Adjacent sequences: A376169 A376170 A376171 * A376173 A376174 A376175

Keyword

nonn,easy

Author

Amiram Eldar, Sep 13 2024

Status

approved