A318511 Numerators of the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.

1, 2, 9, 2, 25, 9, 49, 4, 27, 25, 121, 9, 169, 49, 225, 6, 289, 27, 361, 25, 441, 121, 529, 18, -125, 169, 405, 49, 841, 225, 961, 12, 1089, 289, 1225, 27, 1369, 361, 1521, 50, 1681, 441, 1849, 121, 675, 529, 2209, 27, -1029, -125, 2601, 169, 2809, 405, 3025, 98, 3249, 841, 3481, 225, 3721, 961, 1323, 20, 4225, 1089, 4489, 289, 4761, 1225, 5041, 27

Offset

1,2

Comments

No zeros among the first 2^20 terms.

For odd primes p, it seems that a(p) = p^2.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064549(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Prog

(PARI)
up_to = 65537;
A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); };
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};
v318511_12 = DirSqrt(vector(up_to, n, A064549(n)));
A318511(n) = numerator(v318511_12[n]);

Crossrefs

Cf. A064549, A318512 (denominators).

Cf. also A317935.

Sequence in context: A188966 A248433 A091943 * A345299 A276048 A339203

Adjacent sequences: A318508 A318509 A318510 * A318512 A318513 A318514

Keyword

sign,frac

Author

Antti Karttunen, Aug 30 2018

Status

approved