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For odd n: a(n) = 0, and for even n: a(n) = -mu(n), where mu is Moebius function (A008683).
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%I #24 Jul 06 2024 14:02:30

%S 0,1,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,-1,0,0,0,1,0,0,

%T 0,-1,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,-1,0,0,

%U 0,1,0,0,0,1,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0

%N For odd n: a(n) = 0, and for even n: a(n) = -mu(n), where mu is Moebius function (A008683).

%C Sum of Möbius function values computed for terms of 3x+1 trajectory started at n, but excluding mu(n) itself. See _Marc LeBrun_'s comment in A087003.

%H Antti Karttunen, <a href="/A292273/b292273.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%F a(n) = (A000035(n)-1) * A008683(n).

%F a(n) = A087003(n) - A008683(n).

%F Conjecture: a(n) = Re(A008683(n)*(i^n)). - _Mats Granvik_, Jul 06 2024

%o (PARI)

%o A292273(n) = if(n%2, 0, -moebius(n)); \\ After the definition.

%o \\ Implementation following the Collatz-interpretation:

%o A006370(n) = if(n%2, 3*n+1, n/2); \\ This function from _Michael B. Porter_, May 29 2010

%o A087003(n) = { my(s=1); while(n>1, s += moebius(n); n = A006370(n)); (s); };

%o A292273(n) = (A087003(n)-moebius(n));

%o \\ Or more directly as:

%o A292273(n) = { my(s=0); while(n>1, n = A006370(n); s += moebius(n)); (s); };

%o (Scheme) (define (A292273 n) (* (- (A000035 n) 1) (A008683 n)))

%Y Cf. A000035, A006370, A008683, A014682, A039956 (positions of nonzero terms), A087003.

%K sign,easy

%O 1

%A _Antti Karttunen_, Sep 14 2017