A376170 Powerful numbers whose prime factorization has an even maximum exponent.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 432, 441, 484, 529, 576, 625, 648, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2000, 2025, 2116, 2209

Offset: 1

Author

Amiram Eldar, Sep 13 2024

Keywords

• nonn
• easy

Comments

Powerful numbers k such that A051903(k) is even.

Equivalently, numbers whose prime factorization exponents are all larger than 1 and their maximum is even. The maximum exponent in the prime factorization of 1 is considered to be A051903(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

Complement of A376171 within A001694.

Intersection of A001694 and A368714.

A000290 \ {0} is a subsequence.

Cf. A051903.

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[Max[FactorInteger[#][[;; , 2]]]] &]; seq[10^4]

PARI

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

Formula

Sum_{n>=1} 1/a(n) = Sum_{k>=2} (-1)^k * s(k) = 1.65243302848832032223..., where s(k) = Product_{p prime} (1 + Sum_{i=2..k} 1/p^i).