A335283 a(n) = 1 + Sum_{d|n, n/d odd, d < n} a(d).

1, 1, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 6, 1, 2, 4, 2, 2, 6, 2, 2, 2, 4, 2, 8, 2, 2, 6, 2, 1, 6, 2, 6, 4, 2, 2, 6, 2, 2, 6, 2, 2, 16, 2, 2, 2, 4, 4, 6, 2, 2, 8, 6, 2, 6, 2, 2, 6, 2, 2, 16, 1, 6, 6, 2, 2, 6, 6, 2, 4, 2, 2, 16, 2, 6, 6, 2, 2, 16, 2, 2, 6, 6, 2, 6, 2, 2, 16, 6, 2, 6, 2, 6, 2, 2, 4, 16, 4, 2, 6, 2, 2, 26

Offset: 1

Author

Ilya Gutkovskiy, May 30 2020

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000079 (positions of 1's), A038550 (positions of 2's), A067824, A074206, A209229, A307779, A335062.

Programs

Mathematica

a[n_] := 1 + Sum[If[d < n && OddQ[n/d], a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 85}]
terms = 85; A[] = 0; Do[A[x] = x/(1 - x) + Sum[A[x^(2 k - 1)], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

PARI

up_to = 20000;
A335283lista(up_to) = {my(v = vector(up_to)); for(n=1, up_to, v[n] = 1 + sumdiv(n, d, if(dA335283lista(up_to);
A335283(n) = v335283[n]; \\ Antti Karttunen, Dec 09 2021

Formula

G.f. A(x) satisfies: A(x) = x / (1 - x) + Sum_{k>=2} A(x^(2*k-1)).

G.f.: x / (1 - x) + Sum_{n>=1} a(n) * x^(3*n) / (1 - x^(2*n)).

a(1) = 1; a(2*n) = a(n), a(2*n+1) = 2 * A074206(2*n+1).

Extensions

More terms from Antti Karttunen, Dec 09 2021