A318318 Denominators of rational valued sequence whose Dirichlet convolution with itself yields A173557.

1, 2, 1, 8, 1, 2, 1, 16, 2, 1, 1, 8, 1, 2, 1, 128, 1, 4, 1, 4, 1, 2, 1, 16, 1, 1, 2, 8, 1, 1, 1, 256, 1, 1, 1, 16, 1, 2, 1, 8, 1, 2, 1, 8, 1, 2, 1, 128, 2, 1, 1, 4, 1, 4, 1, 16, 1, 1, 1, 4, 1, 2, 2, 1024, 1, 2, 1, 1, 1, 1, 1, 32, 1, 1, 1, 8, 1, 1, 1, 64, 8, 1, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 4, 2, 1, 1, 1, 1, 8, 1

Offset: 1

Author

Antti Karttunen, Aug 24 2018

Keywords

• nonn
• frac

Comments

Not multiplicative because A318317 contains zeros.

Differs from A317926 at n = 200, 400, 600, 800, 900, 1200, 1400, 1600, 1800, 2200, 2400, 2700, 2800, 3200, 3600, 3800, 4050, 4200, 4400, 4600, 4800, 4900, 5200, ..., which seem to be a subsequence of positions of zeros in A318317.

Here a(200) = 1, while A317926(200) = 2.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A173557, A318317 (numerators).

Cf. also A317926.

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[Denominator[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(nA173557)); 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)));
A318318(n) = denominator(v318317_18[n]);

Formula

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