A318317 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A173557.

1, 1, 1, 3, 2, 1, 3, 5, 1, 1, 5, 3, 6, 3, 2, 35, 8, 1, 9, 3, 3, 5, 11, 5, 0, 3, 1, 9, 14, 1, 15, 63, 5, 4, 6, 3, 18, 9, 6, 5, 20, 3, 21, 15, 1, 11, 23, 35, -3, 0, 8, 9, 26, 1, 10, 15, 9, 7, 29, 3, 30, 15, 3, 231, 12, 5, 33, 3, 11, 3, 35, 5, 36, 9, 0, 27, 15, 3, 39, 35, 3, 10, 41, 9, 16, 21, 14, 25, 44, 1, 18, 33, 15, 23, 18, 63, 48, -3, 5, 0, 50, 4, 51, 15, 6

Offset: 1

Author

Antti Karttunen, Aug 24 2018

Keywords

• sign
• frac

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A173557, A318318 (denominators).

Cf. also A317925, A317935.

Programs

Mathematica

f[1] = 1; f[n_] := f[n] = 1/2 (Module[{fac = FactorInteger[n]}, If[n == 1, 1, Product[fac[[i, 1]] - 1, {i, Length[fac]}]]] - 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 = 16384;
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
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&&dA317937.
v318317_18 = DirSqrt(vector(up_to, n, A173557(n)));
A318317(n) = numerator(v318317_18[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A173557(n) - Sum_{d|n, d>1, d 1.

From Vaclav Kotesovec, May 10 2025: (Start)

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

Sum_{k=1..n} A318317(k) / A318318(k) ~ n^2 * sqrt(f(2)/(4*Pi*log(n))) * (1 + (1 - gamma - f'(2)/f(2) + 6*zeta'(2)/Pi^2) / (4*log(n))), where

f(2) = A307868 = Product_{p prime} (1 - 2/(p*(p+1))) = 0.471680613612997868...

f'(2)/f(2) = Sum_{p prime} 2*p*log(p) / ((p+1)*(p^2+p-2)) = 0.7254208328519472161058521308839896283514823... and gamma is the Euler-Mascheroni constant A001620. (End)