A369564 Powerful numbers whose prime factors are all of the form 4*k + 3.

1, 9, 27, 49, 81, 121, 243, 343, 361, 441, 529, 729, 961, 1089, 1323, 1331, 1849, 2187, 2209, 2401, 3087, 3249, 3267, 3481, 3969, 4489, 4761, 5041, 5929, 6241, 6561, 6859, 6889, 8649, 9261, 9747, 9801, 10609, 11449, 11907, 11979, 12167, 14283, 14641, 16129, 16641

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 A004614.

Similar sequence: A352492, A369563, A369565, A369566.

Cf. A002145, A013661, A175647, A334426.

Programs

Mathematica

q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 4] == 3 && Last[#] > 1 &]; Select[Range[20000], q]

PARI

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%4 != 3 || f[i, 2] == 1, return(0))); 1;}

Formula

Sum_{n>=1} 1/a(n) = Product_{primes p == 3 (mod 4)} (1 + 1/(p*(p-1))) = 3*A013661*A334426/(4*A175647) = 1.2161513254... .