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A083239
First order recursion: a(0) = 1; a(n) = phi(n) - a(n-1) = A000010(n) - a(n-1).
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
0,6
COMMENTS
Provides interesting decomposition: phi(n) = u+w, where u and w consecutive terms of this sequence. Depends also on initial value.
LINKS
FORMULA
a(n) + a(n-1) = A000010(n).
a(n) = (-1)^n * (1 - A068773(n)) for n >= 1. - Amiram Eldar, Mar 05 2024
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}]
PROG
(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
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Labos Elemer, Apr 23 2003
EXTENSIONS
a(0)=1 prepended by R. J. Mathar, Jun 20 2021
STATUS
approved