A096433 - OEIS

%I #30 Oct 26 2020 16:06:39

%S 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,

%T -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,

%U 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

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

%H Robert Israel, <a href="/A096433/b096433.txt">Table of n, a(n) for n = 1..10000</a>

%H Hamed Mousavi and Maxie D. Schmidt, <a href="https://arxiv.org/abs/1810.08373">Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums</a>, arXiv:1810.08373 [math.NT], 2018. See Remark 2.2, pp. 6-7.

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

%e 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.

%p A:= Vector(100):

%p A[1]:= 1:

%p for n from 2 to 100 do

%p   A[n]:= -convert(A[select(t -> igcd(t,n+1)=1, [$1..n-1])],`+`)

%p od:

%p convert(A,list); # _Robert Israel_, Oct 26 2020

%t 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 *)

%K sign

%O 1,13

%A _Leroy Quet_, Aug 10 2004

%E Edited and extended by _Robert G. Wilson v_, Aug 24 2004