A358276 - OEIS

%I #70 Jul 31 2023 02:25:42

%S 1,-2,3,0,5,-18,7,0,18,-30,11,24,13,-42,45,0,17,-144,19,40,63,-66,23,

%T 0,50,-78,108,56,29,-390,31,0,99,-102,105,360,37,-114,117,0,41,-546,

%U 43,88,360,-138,47,0,98,-400,153,104,53,-1080,165,0,171,-174,59,1080,61,-186,504,0,195,-858,67,136

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

%H Seiichi Manyama, <a href="/A358276/b358276.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n * A308077(n).

%F If p is prime, a(p) = (-1)^(p-1) * p.

%F G.f. A(x) satisfies A(x) = x - Sum_{k>=2} (-1)^k * k * A(x^k).

%p b:= proc(n) option remember; `if`(n<2, n, add(b(n/d)*

%p      (-1)^(d-1), d=numtheory[divisors](n) minus {1}))

%p     end:

%p a:= n-> n*b(n):

%p seq(a(n), n=1..68);  # _Alois P. Heinz_, Mar 30 2023

%t a[1] = 1; a[n_] := a[n] = n * DivisorSum[n, (-1)^(n/# - 1) * a[#]/# &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Jul 31 2023 *)

%o (PARI) a(n) = if (n==1, 1, n*sumdiv(n, d, if (d<n, (-1)^(n/d-1)*a(d)/d))); \\ _Michel Marcus_, Mar 30 2023

%Y Cf. A050369, A055615, A308077, A332793.

%K sign,easy

%O 1,2

%A _Seiichi Manyama_, Mar 30 2023