A308077 G.f. A(x) satisfies: A(x) = x - A(x^2) + A(x^3) - A(x^4) + A(x^5) - A(x^6) + ...

1, -1, 1, 0, 1, -3, 1, 0, 2, -3, 1, 2, 1, -3, 3, 0, 1, -8, 1, 2, 3, -3, 1, 0, 2, -3, 4, 2, 1, -13, 1, 0, 3, -3, 3, 10, 1, -3, 3, 0, 1, -13, 1, 2, 8, -3, 1, 0, 2, -8, 3, 2, 1, -20, 3, 0, 3, -3, 1, 18, 1, -3, 8, 0, 3, -13, 1, 2, 3, -13, 1, -4, 1, -3, 8, 2, 3, -13, 1

Offset: 1

Author

Ilya Gutkovskiy, May 11 2019

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006005 (positions of 1's), A067856, A307776, A347031.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, n, add(a(n/d)*
     (-1)^(d-1), d=numtheory[divisors](n) minus {1}))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 30 2023

Mathematica

terms = 79; A[] = 0; Do[A[x] = x + Sum[(-1)^(k + 1) A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := If[n == 1, n, Sum[If[d < n, (-1)^(n/d + 1) a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 79}]

Formula

a(1) = 1; a(n) = Sum_{d|n, d

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