A351405 a(1) = 1; a(n+1) = Sum_{d|n} 2^(n/d - 1) * a(d).

1, 1, 3, 7, 17, 33, 75, 139, 289, 557, 1119, 2143, 4341, 8437, 16843, 33343, 66573, 132109, 264243, 526387, 1052549, 2101617, 4202031, 8396335, 16792705, 33570193, 67137403, 134248191, 268492033, 536927489, 1073853307, 2147595131, 4295180241, 8590155085

Offset: 1

Author

Ilya Gutkovskiy, Feb 10 2022

Keywords

• nonn

Crossrefs

Cf. A003238, A038046, A054599, A308076.

Programs

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      add(2^((n-1)/d-1)*a(d), d=numtheory[divisors](n-1)))
    end:
seq(a(n), n=1..34);  # Alois P. Heinz, Feb 10 2022

Mathematica

a[1] = 1; a[n_] := a[n] = Sum[2^((n - 1)/d - 1) a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 34}]
nmax = 34; A[] = 0; Do[A[x] = x (1 + Sum[2^(k - 1) A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * ( 1 + A(x) + 2 * A(x^2) + 4 * A(x^3) + ... + 2^(k-1) * A(x^k) + ... ).

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

a(n) ~ 2^(n-1). - Vaclav Kotesovec, Feb 18 2022