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A072270 A partial product representation of A006131 and A072265.

%I A072270 #11 Aug 06 2015 01:31:32
%S A072270 1,1,13,9,101,5,701,49,361,29,31021,33,204101,181,1021,1889,8799541,
%T A072270 233,57746701,1361,41581,7589,2486401661,1633,161532401,49661,
%U A072270 22810681,58241,702418373381,2245,4608956945501,3437249,74991181,2135149,2802699901,75921,1302034904649701,14007941,3219888061,3019201
%N A072270 A partial product representation of A006131 and A072265.
%C A072270 Define f(n) = A006131(n-1) and L(n) = 4*f(n-1)+f(n+1), which implies L(n) = A072265(n), n>1.
%C A072270 For even n, f(n) = product_{d|n} a(d) and for odd n, f(n) = product_{d|n} a(2d).
%C A072270 Taking logarithms defines the sequence a(.) via a Mobius transformation (see A072183).
%C A072270 Writing f(n) and L(n) in terms of Binet formulas leads to a representation as cyclotomic polynomials.
%F A072270 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.
%F A072270 a(p) = L(p) for odd prime p.
%F A072270 a(2p) = f(p) for odd prime p.
%F A072270 a(2^k+1) = L(2^k).
%F A072270 a(3*2^k = L(2^k)-4^k.
%F A072270 L(n) = product_{d|n} a(d) for odd n.
%F A072270 L(n*2^k) = product_{d|n} a(d*2^(k+1)) for k>0 and odd n.
%e A072270 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.
%e A072270 f(9)=a(2)*a(6)*a(18)= 1*5*233 = 1165 for odd n=9.
%e A072270 L(6)=a(4)*a(12) = 9*33 = 297 = 4*f(5)+f(7) = 4*29+181 for even n=6.
%e A072270 L(15)=a(1)*a(3)*a(5)*a(15) = 1*13*101*1021 = 1340573 for odd n=15.
%p A072270 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
%Y A072270 Cf. A006131, A072183.
%K A072270 nonn
%O A072270 1,3
%A A072270 _Miklos Kristof_, Jul 09 2002
%E A072270 Divided argument of Phi by 4; moved comments to formula section - _R. J. Mathar_, Nov 17 2010