A323910 Dirichlet inverse of the deficiency of n, A033879.

1, -1, -2, 0, -4, 4, -6, 0, -1, 6, -10, 2, -12, 8, 10, 0, -16, 1, -18, 2, 14, 12, -22, 4, -3, 14, -2, 2, -28, -16, -30, 0, 22, 18, 26, 4, -36, 20, 26, 4, -40, -24, -42, 2, 4, 24, -46, 8, -5, -1, 34, 2, -52, 0, 42, 4, 38, 30, -58, 2, -60, 32, 6, 0, 50, -40, -66, 2, 46, -40, -70, 12, -72, 38, 2, 2, 62, -48, -78, 8, -4, 42, -82, -2, 66, 44, 58, 4, -88, 2, 74, 2

Offset: 1

Author

Antti Karttunen, Feb 12 2019

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Jon Maiga, Computer-generated formulas for A323910, Sequence Machine.

Crossrefs

Cf. A033879, A323911, A323912, A359549 (parity of terms).

Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.

Programs

Mathematica

b[n_] := 2 n - DivisorSigma[1, n];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

PARI

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA033879(n/d) * a(d).

From Antti Karttunen, Nov 14 2024: (Start)

Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:

a(n) = Sum_{d|n} A046692(d)*A067824(n/d).

a(n) = Sum_{d|n} A055615(d)*A074206(n/d).

a(n) = Sum_{d|n} A023900(d)*A174725(n/d).

a(n) = Sum_{d|n} A008683(d)*A323912(n/d).

a(n) = Sum_{d|n} A191161(d)*A327960(n/d).

a(n) = Sum_{d|n} A328722(d)*A330575(n/d).

a(n) = Sum_{d|n} A345182(d)*A349341(n/d).

a(n) = Sum_{d|n} A346246(d)*A349387(n/d).

a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).

(End)