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

Author

Antti Karttunen, Aug 30 2018

Keywords

• sign
• frac

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
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A064549, A318512 (denominators).

Cf. also A317935.

Programs

Mathematica

f[1] = 1; f[n_] := f[n] = 1/2 (n*Times @@ FactorInteger[n][[All, 1]] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Numerator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 10 2025 *)

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&&dA064549(n)));
A318511(n) = numerator(v318511_12[n]);

Formula

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

From Vaclav Kotesovec, May 10 2025: (Start)

Let f(s) = Product_{p prime} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).

Sum_{k=1..n} A318511(k) / A318512(k) ~ n^3 * sqrt(Pi*f(3)/(54*log(n))) * (1 + (1/3 - gamma/2 - f'(3)/(2*f(3)) - 3*zeta'(2)/Pi^2) / (2*log(n))), where

f(3) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298513355936144...

f'(3)/f(3) = Sum_{p prime} (3*p - 2) * log(p) / (p^3 - 2*p + 1) = 1.469536740824614833203393993450164364663334798759143895712...

and gamma is the Euler-Mascheroni constant A001620. (End)