A361985 a(1) = 1, a(2) = 1; a(n) = n * Sum_{d|n, d < n} (-1)^(n/d) a(d) / d.

1, 1, -3, 6, -5, -3, -7, 24, 0, -5, -11, -18, -13, -7, 15, 96, -17, 0, -19, -30, 21, -11, -23, -72, 0, -13, 0, -42, -29, 15, -31, 384, 33, -17, 35, 0, -37, -19, 39, -120, -41, 21, -43, -66, 0, -23, -47, -288, 0, 0, 51, -78, -53, 0, 55, -168, 57, -29, -59, 90, -61, -31, 0, 1536, 65, 33, -67, -102, 69, 35, -71, 0, -73, -37, 0

Offset: 1

Author

Seiichi Manyama, Apr 02 2023

Keywords

• sign
• mult

Links

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

Crossrefs

Partial sums give A359479.

Cf. A361984, A361986.

Cf. A359484.

Programs

Mathematica

f[p_, e_] := If[e == 1, -p, 0]; f[2, e_] := If[e == 1, 1, 6*4^(e-2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 09 2023 *)

Formula

a(n) is multiplicative with a(2) = 1, a(2^e) = 6*4^(e-2) if e>1. a(p) = -p, a(p^e) = 0 if e>1, p>2.

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