A369565 Powerful numbers whose prime factors are all of the form 3*k + 1.

1, 49, 169, 343, 361, 961, 1369, 1849, 2197, 2401, 3721, 4489, 5329, 6241, 6859, 8281, 9409, 10609, 11881, 16129, 16807, 17689, 19321, 22801, 24649, 26569, 28561, 29791, 32761, 37249, 39601, 44521, 47089, 49729, 50653, 52441, 57967, 58081, 61009, 67081, 73441

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 A004611.

Similar sequence: A352492, A369563, A369564, A369566.

Cf. A002476, A175646, A334477.

Programs

Mathematica

q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 3] == 1 && Last[#] > 1 &]; Select[Range[75000], q]

PARI

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

Formula

Sum_{n>=1} 1/a(n) = Product_{primes p == 1 (mod 3)} (1 + 1/(p*(p-1))) = A175646 * A334477 = 1.0377399555...