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

1,4

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)

Formula

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

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 *)

Prog

(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&&d<n, u[d]*u[n/d], 0)))/2); u};
v318650_aux = DirSqrt(vector(up_to, n, A057521(n)));
A318650(n) = numerator(v318650_aux[n]);

Crossrefs

Cf. A057521, A046644 (denominators).

Cf. also A317935, A318511, A318649.

Sequence in context: A366146 A040228 A040229 * A281571 A040227 A040226

Adjacent sequences: A318647 A318648 A318649 * A318651 A318652 A318653

Keyword

nonn,frac,mult

Author

Antti Karttunen, Aug 31 2018

Status

approved