A072271 A partial product representation of f(n) = A015523(n) and L(n) = A072263(n).

3, 1, 24, 19, 431, 14, 7589, 311, 5559, 241, 2345179, 286, 41223001, 4229, 70051

Offset

1,1

Comments

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

For odd prime p, a(p) = L(p)/3, where L(n) = 5*f(n-1) + f(n+1).

a(1)=3, a(2)=1.

a(2p) = f(p) for odd primes p.

a(2^(k+1)) = L(2^k).

a(3*2^k) = L(2^k) - 5^k.

For odd n, L(n) = Product_{d|n} a(d).

For k > 0 and odd n, L(n*2^k) = Product_{d|n} a(d*2^(k+1)).

Links

Table of n, a(n) for n=1..15.

Formula

Let h = (3+sqrt(29))/2, K(n, x) = n-th cyclotomic polynomial, so that x^n - 1 = Product_{d|n} K(d, x); g(d) is the order of K(d, x).

a(n) = (h-3)^g(n)*K(n, h^2).

Example

f(12) = a(1)*a(2)*a(3)*a(4)*a(6)*a(12) = 3*1*24*19*14*286 = 5477472 for even n;
f(7) = a(2)*a(14) = 1*4229 = 4229 for odd n.
L(6) = a(4)*a(12) = 19*286 = 5434 = 5*f(5) + f(7) = 5*241 + 4229 for even n;
L(15) = a(1)*a(3)*a(5)*a(15) = 3*24*431*70051 = 2173822632 for odd n.

Crossrefs

Cf. A072270, A072280, A072183, A127259, A127607.

Sequence in context: A281377 A138654 A175289 * A363818 A309397 A193472

Adjacent sequences: A072268 A072269 A072270 * A072272 A072273 A072274

Keyword

nonn,uned,more

Author

Miklos Kristof, Jul 09 2002

Status

approved