A369566 Powerful numbers whose prime factors are all of the form 3*k + 2.

1, 4, 8, 16, 25, 32, 64, 100, 121, 125, 128, 200, 256, 289, 400, 484, 500, 512, 529, 625, 800, 841, 968, 1000, 1024, 1156, 1331, 1600, 1681, 1936, 2000, 2048, 2116, 2209, 2312, 2500, 2809, 3025, 3125, 3200, 3364, 3481, 3872, 4000, 4096, 4232, 4624, 4913, 5000

Offset: 1

Author

Amiram Eldar, Jan 26 2024

Keywords

• nonn

Comments

Closed under multiplication.

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 A004612.

Similar sequence: A352492, A369563, A369564, A369565.

Cf. A003627, A333240, A334479.

Programs

Mathematica

q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 3] == 2 && Last[#] > 1 &]; Select[Range[5000], q]

PARI

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%3 != 2 || f[i, 2] == 1, return(0))); 1;}

Formula

Sum_{n>=1} 1/a(n) = Product_{primes p == 2 (mod 3)} (1 + 1/(p*(p-1))) = (9/8) * A333240 * A334479 = 1.6053538210...