A380635 a(1) = 1; a(n+1) = Sum_{d^2|n} a(n/d^2).

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 7, 7, 8, 8, 10, 10, 10, 10, 12, 13, 13, 14, 16, 16, 16, 16, 19, 19, 19, 19, 24, 24, 24, 24, 28, 28, 28, 28, 32, 34, 34, 34, 39, 40, 41, 41, 46, 46, 48, 48, 53, 53, 53, 53, 58, 58, 58, 60, 67, 67, 67, 67, 74, 74, 74, 74, 84, 84, 84, 85, 93, 93, 93, 93

Offset: 1

Author

Ilya Gutkovskiy, Jan 28 2025

Keywords

• nonn

Crossrefs

Cf. A003238, A076752, A167865, A307779, A317240.

Programs

Maple

a:= proc(n) option remember; uses numtheory; `if`(n=1, 1,
      add(`if`(issqr(d), a((n-1)/d), 0), d=divisors(n-1)))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 28 2025

Mathematica

a[1] = 1; a[n_] := a[n] = DivisorSum[n - 1, a[(n - 1)/#] &, IntegerQ[Sqrt[#]] &]; Table[a[n], {n, 1, 80}]
nmax = 80; A[] = 0; Do[A[x] = x (1 + Sum[A[x^(k^2)], {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) + A(x^4) + A(x^9) + ... + A(x^(k^2)) + ...).