A072271 - OEIS

%I #15 Sep 19 2024 18:48:02

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

%T 95471,12736968311,5309,223887209309,88321,21607111,1306469,

%U 69176042380099,94846,2821250547551,22964761,160204320879,27289081,375703599163598591,119641

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

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

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

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

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

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

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

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

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

%F a(n) = (h-3)^g(n) * K(n, h^2/5) for n > 2 where h = (3+sqrt(29))/2, Phi(n, x) = n-th cyclotomic polynomial and g(n) is the order of Phi(n, x).

%e f(12) = a(1)*a(2)*a(3)*a(4)*a(6)*a(12) = 3*1*24*19*14*286 = 5477472 for even n;

%e f(7) = a(2)*a(14) = 1*4229 = 4229 for odd n.

%e L(6) = a(4)*a(12) = 19*286 = 5434 = 5*f(5) + f(7) = 5*241 + 4229 for even n;

%e L(15) = a(1)*a(3)*a(5)*a(15) = 3*24*431*70051 = 2173822632 for odd n.

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

%K nonn

%O 1,1

%A _Miklos Kristof_, Jul 09 2002

%E More terms and entry revised by _Sean A. Irvine_, Sep 19 2024