A374458 Squares of exponentially odd numbers (A268335).

1, 4, 9, 25, 36, 49, 64, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 576, 676, 729, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489

Offset

1,2

Comments

Numbers whose exponents in their prime factorization are all congruent to 2 (mod 4).

Links

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

Formula

a(n) = A000290(A268335(n)) = A268335(n)^2.

Sum_{n>=1} 1/a(n) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.54211628314015874165... .

Sum_{n>=1} 1/a(n)^s = zeta(4*s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(4*s)), for s > 1/2.

Mathematica

Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]^2

Prog

(PARI) is(k) = issquare(k) && if(k == 1, 1, my(e = factor(k)[, 2]); for(i = 1, #e, if(e[i] % 4 != 2, return(0))); 1);

Crossrefs

Cf. A000290, A016825, A013662, A268335.

Similar sequences: A030140, A062503, A085986, A153158, A374291.

Sequence in context: A365003 A367406 A030140 * A325240 A355058 A153158

Adjacent sequences: A374455 A374456 A374457 * A374459 A374460 A374461

Keyword

nonn,easy

Author

Amiram Eldar, Jul 09 2024

Status

approved