A072270 -id:A072270 - cp's OEIS frontend

A127607 Sequence arising from the factorization of F(n)= A083099(n) and L(n)= A127226(n).

2, 1, 22, 16, 316, 10, 4264, 184, 2584, 124, 756064, 148, 10050496, 1624, 19216, 31264, 1775616256, 2152, 23600633344, 25936, 3343936, 285856, 4169384372224, 29968, 175371467776, 3798976, 12957013504, 4580416

Offset: 1

Miklos Kristof, Apr 03 2007

nonn

			F(12)=a(1)*a(2)*a(3)*a(4)*a(6)*a(12)=2*1*22*16*10*148=1041920
F(9)=a(2)*a(6)*a(18)= 1*10*2152=21520
L(12)=a(8)*a(24)=184*29968=5514112
L(21)=a(1)*a(3)*a(7)*a(21)=2*22*4264*3343936=627375896576

Cf. A072270, A083099, A127226.

with(numtheory): a[1]:=2:a[2]:=1:for n from 3 to 60 do a[n]:=round(evalf((sqrt(7)-1)^degree(cyclotomic(n, x), x) *cyclotomic(n, (4+sqrt(7))/3), 30)) od: seq(a[n], n=1..60);

a(n)= (sqrt(7)-1)^degree(cyclotomic(n,x),x)*cyclotomic(n,(4+sqrt(7)/3) L(n)=6*F(n-1)+F(n+1) F(2n)=Product(d|2n) a(d), F(2n+1)=Product(d|2n+1) a(2d). L(2n+1)=Product(d|2n+1, a(d)), for k>0: L(2^k*(2n+1))=Product(d|2n+1, a(2^(k+1)*d)). for odd prime p, a(p)=L(p)/2, a(2p)=f(p) a(1)=2, a(2)=1; a(2^(k+1))=L(2^k);

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

Original entry on oeis.org

3, 1, 24, 19, 431, 14, 7589, 311, 5559, 241, 2345179, 286, 41223001, 4229, 70051, 95471, 12736968311, 5309, 223887209309, 88321, 21607111, 1306469, 69176042380099, 94846, 2821250547551, 22964761, 160204320879, 27289081, 375703599163598591, 119641

Offset: 1

Miklos Kristof, Jul 09 2002

nonn

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)).

			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.

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

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).

More terms and entry revised by Sean A. Irvine, Sep 19 2024

cp's OEIS Frontend

A127607 Sequence arising from the factorization of F(n)= A083099(n) and L(n)= A127226(n).

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