1,3

For even n, F(n) = Product(d|n)a(d) and for odd n, F(n) = Product(d|n)a(2d).

For odd non-composite n, a(n)=L(n), where L(n) is the n-th Lucas number. a(2)=1. Also a(2p)=F(p) for odd primes.

For even n, F(n)=Product(d|n)a(d). So for even n, log(F(n))=Sum(d|n)log(a(d)). For odd n, L(n)=Product(d|n)a(d). So for odd n, log(L(n))=Sum(d|n)log(a(d)). So we can use the Moebius-Transformation for getting a(n).

Let h=(1+sqrt(5))/2, K(n, x) = n-th cyclotomic polynomial, so that x^n-1= Product(d|n)K(d, x); f(d) is the order of K(d, x). a(n)=(h-1)^f(n)*K(n, h+1)

For odd n: log(a(n))=Sum(d|n)mu(n/d)*log(L(d)). For even n:log(a(n))=Sum(d|n, d even)mu(n/d)*log(F(d))+Sum(d|n, d odd)mu(n/d)*log(L(d))

F(12)=a(1)*a(2)*a(3)*a(4)*a(6)*a(12) = 1*1*4*3*2*6 = 144 for even n, F(15)=a(2)*a(6)*a(10)*a(30) = 1*2*5*61 = 610 for odd n.

For even n: log(a(12))=mu(6)*log(F(2))+mu(3)*log(F(4))+mu(2)*log(F(6))+ +mu(1)*log(F12))+mu(12)*log(L(1))+mu(4)*log(L(3))=0-log(3)-log(8)+log(144)+0+0 = log(144/3/8) = log(6): a(12)=6 For odd n: log(a(15))=mu(15)*log(L(1))+mu(5)*log(L(3))+mu(3)*log(L(5))+ +mu(1)*log(L(15))=0-log(4)-log(11)+log(1364)=log(1364/4/11))=log(31) so a(15)=31.

F[n_] := Fibonacci[n]; L[n_] := F[n + 1] + F[n - 1]; a[2] = 1; a[n_] := a[n] = If[ PrimeQ[n] || n == 1, L[n], If[ PrimeQ[n/2] && OddQ[n/2], F[n/2], If[ EvenQ[n], F[n]/b[n], a[2n] = F[n]/b[n]; F[2n]/c[2n]]]]; b[n_] := (d = Delete[ Divisors[n], -1]; p = 1; k = 1; l = Length[d]; While[k < l + 1, p = p*If[EvenQ[n], a[ d[[k]]], a[ 2d[[k]]]]; k++ ]; p); c[n_] := (d = Delete[Divisors[n], -2]; p = 1; k = 1; l = Length[d]; While[k < l + 1, p = p*a[ d[[k]]]; k++ ]; p); Table[ a[n], {n, 1, 50}]

Sequence in context: A103252 A065763 A100492 this_sequence A005013 A086564 A080777

Adjacent sequences: A072180 A072181 A072182 this_sequence A072184 A072185 A072186

nonn

Miklos Kristof (kristmikl(AT)freemail.hu), Jul 01 2002

Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com) and Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 02 2002