A318650 Numerators of the sequence whose Dirichlet convolution with itself yields A057521, the powerful part of n.

1, 1, 1, 15, 1, 1, 1, 49, 35, 1, 1, 15, 1, 1, 1, 603, 1, 35, 1, 15, 1, 1, 1, 49, 99, 1, 181, 15, 1, 1, 1, 2023, 1, 1, 1, 525, 1, 1, 1, 49, 1, 1, 1, 15, 35, 1, 1, 603, 195, 99, 1, 15, 1, 181, 1, 49, 1, 1, 1, 15, 1, 1, 35, 14875, 1, 1, 1, 15, 1, 1, 1, 1715, 1, 1, 99, 15, 1, 1, 1, 603, 3235, 1, 1, 15, 1, 1, 1, 49, 1, 35, 1, 15, 1, 1, 1, 2023, 1

Offset: 1

Author

Antti Karttunen, Aug 31 2018

Keywords

• nonn
• frac
• mult

Comments

Multiplicative because A046644 and A057521 are.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Vaclav Kotesovec, Graph - the asymptotic ratio (10^8 terms)

Crossrefs

Cf. A057521, A046644 (denominators).

Cf. also A317935, A318511, A318649.

Programs

Mathematica

ff[p_, e_] := If[e > 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ ff @@@ FactorInteger[n]; f[1] = 1; f[n_] := f[n] = 1/2 (a[n] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Numerator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 11 2025 *)

PARI

up_to = 65537;
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
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&&dA057521(n)));
A318650(n) = numerator(v318650_aux[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A057521(n) - Sum_{d|n, d>1, d 1.

From Vaclav Kotesovec, May 10 2025, simplified May 11 2025: (Start)

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

Sum_{k=1..n} A318650(k) / A046644(k) ~ n^(3/2) * sqrt(2*f(3/2)/(9*Pi*log(n))) * (1 + (2/3 - gamma - f'(3/2)/(2*f(3/2))) / (2*log(n))), where

f(3/2) = Product_{p prime} (1 + 2/p^(3/2) - 1/p^(5/2)) = A328013 = 3.51955505841710664719752940369857817...

f'(3/2)/f(3/2) = Sum_{p prime} (4*p - 3) * log(p) / (1 - 2*p - p^(5/2)) = -3.90914718020692131140714384422938370058563543737256496...

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