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A275205 Partial sums of the Dirichlet inverse of the Euler totient function.

%I A275205 #17 Jan 29 2019 12:13:01
%S A275205 1,0,-2,-3,-7,-5,-11,-12,-14,-10,-20,-18,-30,-24,-16,-17,-33,-31,-49,
%T A275205 -45,-33,-23,-45,-43,-47,-35,-37,-31,-59,-67,-97,-98,-78,-62,-38,-36,
%U A275205 -72,-54,-30,-26,-66,-78,-120,-110,-102,-80,-126,-124,-130
%N A275205 Partial sums of the Dirichlet inverse of the Euler totient function.
%C A275205 The matrix A191898 seen as a 2n X 2n block matrix A = {{a, b},{c, d}} where a, b, c and d are n X n matrices, A191898 has the property that the sum: 1 - a - d = trace(A191898). This is demonstrated by the second Mathematica program below.
%H A275205 Vaclav Kotesovec, <a href="/A275205/b275205.txt">Table of n, a(n) for n = 1..10000</a>
%F A275205 a(n) = Trace of matrix A191898.
%F A275205 a(n) = Sum_{k=1..n} A023900(k).
%F A275205 a(n) = 2 - Sum_{m=1..n} Sum_{k=1..n} A191898(m,k).
%F A275205 a(n) = 1 - Sum_{m=1..n} Sum_{k=1..n} A191898(m,k) - Sum_{m=n+1..2*n} Sum_{k=n+1..2*n} A191898(m,k).
%F A275205 a(n) = Sum_{k=1..n} k * mu(k) * floor(n/k), where mu(k) is the Moebius function. - _Daniel Suteu_, Jun 11 2018
%p A275205 a_list := len -> ListTools:-PartialSums([seq(mul(1-i, i=numtheory:-factorset(k)), k=1..len)]): a_list(49); # _Peter Luschny_, Jul 20 2016
%t A275205 Clear[a, n, d]; a[n_] := If[n < 1, 0, Sum[d MoebiusMu@d, {d, Divisors[n]}]]; Accumulate[Table[a[n], {n, 1, 49}]] (* After Michael Somos in A023900 *)
%t A275205 Clear[jj];
%t A275205 jj = 49;
%t A275205 Table[
%t A275205 Clear[nn, A, B, AB, n, k];
%t A275205 nn = 2*ii;
%t A275205 A = Table[
%t A275205    Table[If[Mod[n, k] == 0, Sqrt[k], 0], {k, 1, nn}], {n, 1, nn}];
%t A275205 B = Table[
%t A275205    Table[If[Mod[k, n] == 0, MoebiusMu[n]*Sqrt[n], 0], {k, 1, nn}], {n,
%t A275205      1, nn}]; MatrixForm[AB = A.B];
%t A275205 a = Table[Table[AB[[n, k]], {k, 1, nn/2}], {n, 1, nn/2}];
%t A275205 d = Table[Table[AB[[n, k]], {k, nn/2 + 1, nn}], {n, nn/2 + 1, nn}];
%t A275205 1 - Total[Total[a + d]], {ii, 1, jj}]
%Y A275205 Cf. A023900, A191898.
%K A275205 sign
%O A275205 1,3
%A A275205 _Mats Granvik_, Jul 19 2016