Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #15 May 10 2023 04:31:33
%S 1,1,-3,6,-5,-3,-7,24,0,-5,-11,-18,-13,-7,15,96,-17,0,-19,-30,21,-11,
%T -23,-72,0,-13,0,-42,-29,15,-31,384,33,-17,35,0,-37,-19,39,-120,-41,
%U 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
%N a(1) = 1, a(2) = 1; a(n) = n * Sum_{d|n, d < n} (-1)^(n/d) a(d) / d.
%H Seiichi Manyama, <a href="/A361985/b361985.txt">Table of n, a(n) for n = 1..10000</a>
%F 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.
%F G.f. A(x) satisfies -x * (1 - x) = Sum_{k>=1} (-1)^k * k * A(x^k).
%t 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 *)
%Y Partial sums give A359479.
%Y Cf. A361984, A361986.
%Y Cf. A359484.
%K sign,mult
%O 1,3
%A _Seiichi Manyama_, Apr 02 2023