A377818 Powerful numbers that have a single even exponent in their prime factorization.

4, 9, 16, 25, 49, 64, 72, 81, 108, 121, 169, 200, 256, 288, 289, 361, 392, 432, 500, 529, 625, 648, 675, 729, 800, 841, 961, 968, 972, 1024, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1681, 1728, 1849, 2000, 2209, 2312, 2401, 2592, 2809, 2888, 3087, 3200, 3267, 3481

Offset: 1

Author

Amiram Eldar, Nov 09 2024

Keywords

• nonn
• easy

Comments

Each term can be represented in a unique way as m * p^(2*k), k >= 1, where m is a cubefull exponentially odd number (A335988) and p is a prime that does not divide m.

Powerful numbers k such that A350388(k) is a prime power with an even positive exponent (A056798 \ {1}).

Links

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

Crossrefs

Intersection of A001694 and A377816.

Subsequence of A377819.

Cf. A056798, A065487, A335988, A350388.

Programs

Mathematica

With[{max = 3500}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], _?EvenQ] == 1 &]]

PARI

is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> !(x%2), e) == 1);

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) * Sum_{p prime} (p/(p^3-p+1)) = 0.61399274770712398109... .