A072270 A partial product representation of A006131 and A072265.

1, 1, 13, 9, 101, 5, 701, 49, 361, 29, 31021, 33, 204101, 181, 1021, 1889, 8799541, 233, 57746701, 1361, 41581, 7589, 2486401661, 1633, 161532401, 49661, 22810681, 58241, 702418373381, 2245, 4608956945501, 3437249, 74991181, 2135149, 2802699901, 75921, 1302034904649701, 14007941, 3219888061, 3019201

Offset: 1

Author

Miklos Kristof, Jul 09 2002

Keywords

• nonn

Comments

Define f(n) = A006131(n-1) and L(n) = 4*f(n-1)+f(n+1), which implies L(n) = A072265(n), n>1.

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

Taking logarithms defines the sequence a(.) via a Mobius transformation (see A072183).

Writing f(n) and L(n) in terms of Binet formulas leads to a representation as cyclotomic polynomials.

Examples

			f(12)=a(1)*a(2)*a(3)*a(4)*a(6)*a(12) = 1*1*13*9*5*33 = 19305 for even n=12.
f(9)=a(2)*a(6)*a(18)= 1*5*233 = 1165 for odd n=9.
L(6)=a(4)*a(12) = 9*33 = 297 = 4*f(5)+f(7) = 4*29+181 for even n=6.
L(15)=a(1)*a(3)*a(5)*a(15) = 1*13*101*1021 = 1340573 for odd n=15.
		

Crossrefs

Cf. A006131, A072183.

Programs

Maple

A072270 := proc(n) if n <=2 then 1; else h := (1+sqrt(17))/2 ; cy := numtheory[cyclotomic](n,x) ; g := degree(cy) ; (h-1)^g*subs(x=h^2/4,cy) ; expand(%) ; end if; end proc: # R. J. Mathar, Nov 17 2010

Formula

Let h=(1+sqrt(17))/2, Phi(n, x) = n-th cyclotomic polynomial, so x^n-1= product_{d|n} Phi(d, x), and let g(d) be the order of Phi(d, x). Then a(n)=(h-1)^g(n)*Phi(n, h^2/4), n>2.

a(p) = L(p) for odd prime p.

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

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

a(3*2^k = L(2^k)-4^k.

L(n) = product_{d|n} a(d) for odd n.

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

Extensions

Divided argument of Phi by 4; moved comments to formula section - R. J. Mathar, Nov 17 2010