A318653 Numerators of the sequence whose Dirichlet convolution with itself yields A007947, the squarefree kernel of n.

1, 1, 3, 1, 5, 3, 7, 1, 3, 5, 11, 3, 13, 7, 15, 3, 17, 3, 19, 5, 21, 11, 23, 3, -5, 13, 15, 7, 29, 15, 31, 3, 33, 17, 35, 3, 37, 19, 39, 5, 41, 21, 43, 11, 15, 23, 47, 9, -21, -5, 51, 13, 53, 15, 55, 7, 57, 29, 59, 15, 61, 31, 21, 5, 65, 33, 67, 17, 69, 35, 71, 3, 73, 37, -15, 19, 77, 39, 79, 15, 3, 41, 83, 21, 85, 43, 87, 11, 89, 15

Offset: 1

Author

Antti Karttunen, Aug 31 2018

Keywords

• sign
• frac
• mult

Comments

No zeros among the first 2^20 terms.

Links

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

Crossrefs

Cf. A007947, A299150 (denominators).

Cf. also A317935, A318511, A318512, A318649.

Programs

Mathematica

rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); f[1] = 1; f[n_] := f[n] = (rad[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; a[n_] := Numerator [f[n]]; Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

PARI

up_to = 65537;
A007947(n) = factorback(factorint(n)[, 1]);
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&&dA007947(n)));
A318653(n) = numerator(v318653_aux[n]);
for(n=1, 100, print1(numerator(direuler(p=2, n, ((1 + p*X - X)/(1 - X))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 08 2025

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A007947(n) - Sum_{d|n, d>1, d 1.

From Vaclav Kotesovec, May 08 2025: (Start)

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

Sum_{k=1..n} A318653(k)/A299150(k) ~ n^2 * sqrt(Pi*f(2)/(24*log(n))) * (1 - (gamma - 1 + f'(2)/f(2) + 6*zeta'(2)/Pi^2) / (4*log(n))), where

f(2) = A065464 = Product_{p prime} (1 - 2/p^2 + 1/p^3) = 0.4282495056770944402187657075818235461212985133559361440319...

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

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