A083238 First order recursion: a(0)=1; a(n) = sigma(1,n) - a(n-1).

1, 0, 3, 1, 6, 0, 12, -4, 19, -6, 24, -12, 40, -26, 50, -26, 57, -39, 78, -58, 100, -68, 104, -80, 140, -109, 151, -111, 167, -137, 209, -177, 240, -192, 246, -198, 289, -251, 311, -255, 345, -303, 399, -355, 439, -361, 433, -385, 509, -452, 545, -473, 571, -517, 637, -565, 685, -605, 695, -635, 803, -741, 837, -733, 860

Offset: 0

Author

Labos Elemer, Apr 23 2003

Keywords

• sign

Comments

Provide interesting decomposition: sigma(n)=u+w, where u and w consecutive terms of this sequence; this depends also on initial value.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000203, A083236, A083237.

Programs

Mathematica

f[x_] := DivisorSigma[1, x]-f[x-1] f[0]=1; Table[f[w], {w, 1, 100}]
nxt[{n_,a_}]:={n+1,DivisorSigma[1,n+1]-a}; NestList[nxt,{0,1},70][[;;,2]] (* Harvey P. Dale, May 10 2024 *)

PARI

lista(nn) = {my(last = 1, v=vector(nn)); for (n=1, nn, v[n] = sigma(n) - last; last = v[n]; ); concat(1, v); } \\ Michel Marcus, Mar 28 2020

Formula

It follows that a(n)+a(n-1) = A000203(n).