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

1,2

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

Formula

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

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 *)

Prog

(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&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v318317_18 = DirSqrt(vector(up_to, n, A173557(n)));
A318318(n) = denominator(v318317_18[n]);

Crossrefs

Cf. A173557, A318317 (numerators).

Cf. also A317926.

Sequence in context: A011327 A318454 A317926 * A318314 A230369 A341406

Adjacent sequences: A318315 A318316 A318317 * A318319 A318320 A318321

Keyword

nonn,frac

Author

Antti Karttunen, Aug 24 2018

Status

approved