A178412 a(n) is defined recursively as the Sum{d|n} ((-1)^(n/d))*a(d) = 1, with a(1) = a(2) = 1.

1, 1, -2, 1, -2, -3, -2, 2, 0, -3, -2, -5, -2, -3, 2, 4, -2, 0, -2, -5, 2, -3, -2, -10, 0, -3, 0, -5, -2, 3, -2, 8, 2, -3, 2, 0, -2, -3, 2, -10, -2, 3, -2, -5, 0, -3, -2, -20, 0, 0, 2, -5, -2, 0, 2, -10, 2, -3, -2, 5, -2, -3, 0, 16, 2, 3, -2, -5, 2, 3, -2, 0, -2, -3, 0, -5, 2, 3, -2, -20, 0, -3, -2, 5, 2, -3, 2, -10, -2, 0, 2, -5, 2, -3, 2, -40, -2, 0, 0, 0, -2, 3

Offset

1,3

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

If n>=3 is odd squarefree (A056911), then a(n)=2; if n>=6 is even squarefree (A039956), then a(n)=3; if n>=4 has the form (2^k)*m, where k>=2 and m is odd squarefree, then a(n)=n/4 in case of m=1 and a(n)=5*2^(k-2) in case of m>1; if an odd square divides n, then a(n)=0. A generalization. Let A(n)=|B(n)|, where B(1)=f, B(2)=g and, for n>=3, B(n)is defined by the recursion: Sum{d|n}((-1)^(n/d))*B(d)=h. Then we have:if n>=3 is odd squarefree,then A(n)=f+h; if n>=6 is even squarefree,then A(n)=g+2*h; if n>=4 has the form (2^k)*m, where k>=2 and m is odd squarefree, then A(n)=n/4 in case of m=1 and A(n)=(f+g+3*h)*2^(k-2) in case of m>1; if an odd square divides n, then A(n)=0.

Example

For n=3, we have Sum{d=1,3}((-1)^(3/d))*b(d)=-1-b(3)=1. Thus b(3)-2 and a(3)=2.

Mathematica

a[1] = a[2] = 1; a[n_] := a[n] = Block[{d = Most@ Divisors@ n}, -1 + Plus @@ (((-1)^(n/#)) a[ # ] & /@ d)]; Array[a, 102] (* Robert G. Wilson v, Aug 26 2010 *)

Prog

(PARI)
up_to = 65537;
A178412list(up_to) = { my(u=vector(up_to)); u[1] = u[2] = 1; for(n=3, up_to, u[n] = sumdiv(n, d, if(d<n, ((-1)^(n/d))*u[d]))-1); (u); };
v178412 = A178412list(up_to);
A178412(n) = v178412[n]; \\ Antti Karttunen, Sep 27 2018

Crossrefs

Cf. A178411.

Sequence in context: A088062 A248886 A123884 * A182598 A331084 A067694

Adjacent sequences: A178409 A178410 A178411 * A178413 A178414 A178415

Keyword

sign

Author

Vladimir Shevelev, May 27 2010

Extensions

Edited and extended by Robert G. Wilson v, Aug 26 2010

Title changed and terms past a(42) added by Robert G. Wilson v, Aug 26 2010

Status

approved