A178411 - OEIS

%I #12 Sep 22 2017 07:46:23

%S 2,1,-1,4,-1,1,-1,8,0,1,-1,0,-1,1,1,16,-1,0,-1,0,1,1,-1,0,0,1,0,0,-1,

%T -1,-1,32,1,1,1,0,-1,1,1,0,-1,-1,-1,0,0,1,-1,0,0,0,1,0,-1,0,1,0,1,1,

%U -1,0,-1,1,0,64,1,-1,-1,0,1,-1,-1,0,-1,1,0,0,1,-1,-1,0,0,1,-1,0,1,1,1,0,-1,0,1,0,1,1,1,0,-1,0,0,0,-1,-1,-1,0,-1

%N a(1)=2, a(2)=1; for n>=3, a(n) is defined by recursion: Sum_{d|n}((-1)^(n/d))*a(d) = -1.

%C A generalization of the sequence: Let a(1) = m+1, a(2) = 2*m-1, and for n>=3, a(n) is defined by recursion: Sum{d|n}((-1)^(n/d))*a(d) = -m. Then a(n) = mu(n), if n is not power of 2; otherwise, for n>=4, a(n) = m*n.

%H Antti Karttunen, <a href="/A178411/b178411.txt">Table of n, a(n) for n = 1..16384</a>

%F a(1) = 2, a(2) = 1, and for n > 2, if A209229(n) = 1 [n is a power of 2] then a(n) = n, otherwise a(n) = A008683(n), where A008683(n) is Moebius mu function.

%F a(n) = 1 + Sum_{d|n, d<n} ((-1)^(n/d))*a(d). [Description converted from an implicit to an explicit recurrence] - _Antti Karttunen_, Sep 21 2017

%t a[1] = 2; a[2] = 1; a[n_] := a[n] = If[IntegerQ@ Log2@ n, # + 1, MoebiusMu[n]] &@ Sum[((-1)^(n/d)) a[d], {d, Most@ Divisors@ n}]; Array[a, 105] (* _Michael De Vlieger_, Sep 21 2017 *)

%o (PARI) A178411(n) = { my(s=1); if(n<=2,3-n,fordiv(n,d,if(d<n,s+=(((-1)^(n/d))*A178411(d)))); s); }; \\ _Antti Karttunen_, Sep 21 2017

%Y Cf. A008683, A209229.

%K sign

%O 1,1

%A _Vladimir Shevelev_, May 27 2010

%E More terms and editing from _Antti Karttunen_, Sep 21 2017