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

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

Author

Labos Elemer, Apr 23 2003

Keywords

• sign
• easy

Comments

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

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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

Python

# 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

Formula

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

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

Extensions

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