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

Author

Antti Karttunen, Aug 31 2018

Keywords

• nonn
• frac

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A000290, A318512 (denominators).

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

Programs

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&&dA318649(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

Formula

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