A072183 Sequence arising from factorization of the Fibonacci numbers.

1, 1, 4, 3, 11, 2, 29, 7, 19, 5, 199, 6, 521, 13, 31, 47, 3571, 17, 9349, 41, 211, 89, 64079, 46, 15251, 233, 5779, 281, 1149851, 61, 3010349, 2207, 9901, 1597, 64681, 321, 54018521, 4181, 67861, 2161, 370248451, 421, 969323029, 13201, 97921

Offset: 1

Author

Miklos Kristof, Jul 01 2002

Keywords

• nonn

Comments

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

For odd noncomposite 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).

Examples

			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(F(12)) + 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.
		

Programs

Mathematica

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}]

Formula

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)).

Extensions

Edited and extended by Robert G. Wilson v and Vladeta Jovovic, Jul 02 2002