A127332 - OEIS

%I #23 Aug 01 2019 03:56:38

%S 1,2,2,3,3,3,5,4,4,5,9,1,10,8,3,7,15,3,16,2,6,17,21,-6,13,19,11,8,27,

%T -5,27,10,13,28,10,-10,35,31,17,-6,40,-3,40,20,-4,40,44,-18,32,18,26,

%U 23,50,4,21,0,28,54,58,-45,59,53,3,19,24,11,65,37,39,1

%N A126988 * A002321.

%H Michael De Vlieger, <a href="/A127332/b127332.txt">Table of n, a(n) for n = 1..10000</a>

%F M * V where M = A126988 as an infinite lower triangular matrix and V = the Mertens sequence, A002321 as a vector: [1, 0, -1, -1, -2, -1, ...].

%F a(n) = Sum_{q=1..n} c_q(n), where c_q(n) is the Ramanujan's sum function given in A054533. - _Daniel Suteu_, Jun 14 2018

%e a(6) = 3 = 6*1 + 3*0 + 2*(-1) + 0*(-1) + 0*(-2) + 1*(-1), where (6, 3, 2, 0, 0, 1) = row 6 of A126988.

%t Block[{nn = 70, m}, m = Table[Sum[MoebiusMu@k, {k, n}], {n, nn}]; Table[Total@ Array[m[[#]] If[Mod[n, #] == 0, n/#, 0] &, n], {n, nn}]] (* _Michael De Vlieger_, Jun 14 2018 *)

%o (PARI) lista(nn) = {mat = matrix(nn, nn, n, k, if (n % k, 0, n/k)); vec = matrix(nn, 1, n, k, if (k==1, mertens(n), 0)); res = (mat*vec); for (n = 1, nn, print1(res[n, 1], ", "););} \\ _Michel Marcus_, Sep 25 2013

%o (PARI) a(n) = sum(k=1, n, moebius(k / gcd(n, k)) * eulerphi(k) / eulerphi(k / gcd(n, k))); \\ _Daniel Suteu_, Jun 23 2018

%Y Cf. A002321, A054532, A054533, A054534, A054535, A126988.

%K sign

%O 1,2

%A _Gary W. Adamson_, Jan 10 2007

%E Corrected and extended by _Michel Marcus_, Sep 25 2013