A083237 -id:A083237 - cp's OEIS frontend

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

Labos Elemer, Apr 23 2003

sign

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

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

Cf. A000203, A083236, A083237.

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 *)

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

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

A083239 First order recursion: a(0) = 1; a(n) = phi(n) - a(n-1) = A000010(n) - a(n-1).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, -1, 7, -3, 9, -5, 15, -11, 23, -17, 25, -17, 33, -27, 45, -37, 49, -39, 61, -53, 73, -61, 79, -67, 95, -87, 117, -101, 121, -105, 129, -117, 153, -135, 159, -143, 183, -171, 213, -193, 217, -195, 241, -225, 267, -247, 279, -255, 307, -289, 329, -305, 341, -313, 371, -355, 415, -385, 421, -389, 437, -417

Offset: 0

Labos Elemer, Apr 23 2003

Provides interesting decomposition: phi(n) = u+w, where u and w consecutive terms of this sequence. Depends also on initial value.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000010, A068773, A083236, A083237, A083238.

A083239 := proc(n)
    option remember ;
    if n = 0 then
        1 ;
    else
        numtheory[phi](n)-procname(n-1) ;
    end if;
end proc:
seq(A083239(n),n=0..100) ; # R. J. Mathar, Jun 20 2021

a[n_] := a[n] = EulerPhi[n] -a[n-1]; a[0] = 1; Table[a[n], {n, 0, 100}]

# uses programs from A002088 and A049690
def A083239(n): return A002088(n)-(A049690(n>>1)<<1)-1 if n&1 else 1+(A049690(n>>1)<<1)-A002088(n) # Chai Wah Wu, Aug 04 2024

a(n) + a(n-1) = A000010(n).

a(n) = (-1)^n * (1 - A068773(n)) for n >= 1. - Amiram Eldar, Mar 05 2024

a(0)=1 prepended by R. J. Mathar, Jun 20 2021

cp's OEIS Frontend

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

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