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

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).

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]);

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

Author

Antti Karttunen, Aug 31 2018

Status

approved