A096433 a(1) = 1; for n > 1, choose a(n) so that Sum_{1 <= k <= n, gcd(k,n+1)=1} a(k) = 0.

1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 3, -3, -1, 1, -3, 3, 1, -1, 1, -1, -1, -1, 5, -1, -1, -1, -1, 1, -1, 1, -3, 5, -3, 1, 7, -5, -1, -1, -9, 9, 5, 3, 3, -11, -3, 7, 7, 9, -1, -19, -7, 17, 11, 9, -7, -23, 1, -1, -1, 37, 1, -33, -1, -3, -3, 15, 27, -39, -7, 7, -9, 47, -13, -37, 11, 1, -5, 51, -9, -37, 19, 17, -5, -1, 13, -43, -5, -3, 13

Offset

1,13

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Hamed Mousavi and Maxie D. Schmidt, Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums, arXiv:1810.08373 [math.NT], 2018. See Remark 2.2, pp. 6-7.

Formula

a(n) = -Sum_{1 <= k <= n-1, gcd(k, n+1) = 1} a(k).

Example

a(7) = 1 since the positive integers < 8 and coprime to 8 are 1, 3, 5, 7, and thus a(1) + a(3) + a(5) + a(7) = 1 - 1 - 1 + 1 = 0.

Maple

A:= Vector(100):
A[1]:= 1:
for n from 2 to 100 do
  A[n]:= -convert(A[select(t -> igcd(t, n+1)=1, [$1..n-1])], `+`)
od:
convert(A, list); # Robert Israel, Oct 26 2020

Mathematica

a[1] = 1; a[n_] := a[n] = Block[{k = Select[ Range[n - 1], GCD[ #, n + 1] == 1 &]}, -Plus @@ (a /@ k)]; Table[ a[n], {n, 94}] (* Robert G. Wilson v, Aug 24 2004 *)

Crossrefs

Sequence in context: A350617 A337743 A335624 * A084101 A053386 A338696

Adjacent sequences: A096430 A096431 A096432 * A096434 A096435 A096436

Keyword

sign

Author

Leroy Quet, Aug 10 2004

Extensions

Edited and extended by Robert G. Wilson v, Aug 24 2004

Status

approved