A381314 Powerful numbers that have a single exponent in their prime factorization that equals 2.

4, 9, 25, 49, 72, 108, 121, 144, 169, 200, 288, 289, 324, 361, 392, 400, 500, 529, 576, 675, 784, 800, 841, 961, 968, 972, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1600, 1681, 1849, 1936, 2025, 2209, 2304, 2312, 2500, 2704, 2809, 2888, 2916, 3087, 3136, 3200

Offset: 1

Author

Amiram Eldar, Feb 19 2025

Keywords

• nonn
• easy

Comments

Number of the form A036966(m)/p, m >= 2, where p is a prime divisor of A036966(m).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A001694, A377816 and A377818.

Subsequences: A001248, A143610, A179646, A179689, A179699, A189988, A189990, A190106, A190115, A190470, A190471.

Cf. A036966, A065483.

Programs

Mathematica

With[{max = 3200}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], 2] == 1 &]]

PARI

isok(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) = Sum_{p prime}((p-1)/(p^3-p^2+1)) * Product_{p prime} (1 + 1/(p^2*(p-1))) = 0.53045141423939736076... .