A318649 Numerators of the sequence whose Dirichlet convolution with itself yields squares, A000290.

1, 2, 9, 6, 25, 9, 49, 20, 243, 25, 121, 27, 169, 49, 225, 70, 289, 243, 361, 75, 441, 121, 529, 90, 1875, 169, 3645, 147, 841, 225, 961, 252, 1089, 289, 1225, 729, 1369, 361, 1521, 250, 1681, 441, 1849, 363, 6075, 529, 2209, 315, 7203, 1875, 2601, 507, 2809, 3645, 3025, 490, 3249, 841, 3481, 675, 3721, 961, 11907, 924, 4225, 1089

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * ((n^2) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

a(n) = n*A318512(n)*A299149(n)/A299150(n).

Sum_{k=1..n} A318649(k) / A318512(k) ~ n^3/(3*sqrt(Pi*log(n))) * (1 + (1 - 3*gamma/2) / (6*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 09 2025

Prog

(PARI)
up_to = 65537;
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u};
v318649_aux = DirSqrt(vector(up_to, n, (n*n)));
A318649(n) = numerator(v318649_aux[n]);
(PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p^2*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

Crossrefs

Cf. A000290, A318512 (denominators).

Cf. also A046643, A299149, A318511, A318651, A318654 (gives the positions of even terms), A318655 (the 2-adic valuation).

Sequence in context: A068632 A320037 A122664 * A033152 A009306 A324555

Adjacent sequences: A318646 A318647 A318648 * A318650 A318651 A318652

Keyword

nonn,frac

Author

Antti Karttunen, Aug 31 2018

Status

approved