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

1, 0, -2, -3, -7, -5, -11, -12, -14, -10, -20, -18, -30, -24, -16, -17, -33, -31, -49, -45, -33, -23, -45, -43, -47, -35, -37, -31, -59, -67, -97, -98, -78, -62, -38, -36, -72, -54, -30, -26, -66, -78, -120, -110, -102, -80, -126, -124, -130

Offset: 1

Author

Mats Granvik, Jul 19 2016

Keywords

• sign

Comments

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.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A023900, A191898.

Programs

Maple

a_list := len -> ListTools:-PartialSums([seq(mul(1-i, i=numtheory:-factorset(k)), k=1..len)]): a_list(49); # Peter Luschny, Jul 20 2016

Mathematica

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 *)
Clear[jj];
jj = 49;
Table[
Clear[nn, A, B, AB, n, k];
nn = 2*ii;
A = Table[
   Table[If[Mod[n, k] == 0, Sqrt[k], 0], {k, 1, nn}], {n, 1, nn}];
B = Table[
   Table[If[Mod[k, n] == 0, MoebiusMu[n]*Sqrt[n], 0], {k, 1, nn}], {n,
     1, nn}]; MatrixForm[AB = A.B];
a = Table[Table[AB[[n, k]], {k, 1, nn/2}], {n, 1, nn/2}];
d = Table[Table[AB[[n, k]], {k, nn/2 + 1, nn}], {n, nn/2 + 1, nn}];
1 - Total[Total[a + d]], {ii, 1, jj}]

Formula

a(n) = Trace of matrix A191898.

a(n) = Sum_{k=1..n} A023900(k).

a(n) = 2 - Sum_{m=1..n} Sum_{k=1..n} A191898(m,k).

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).

a(n) = Sum_{k=1..n} k * mu(k) * floor(n/k), where mu(k) is the Moebius function. - Daniel Suteu, Jun 11 2018