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A072270 A partial product representation of A006131 and A072265. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..40.

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.

EXAMPLE

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.

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

CROSSREFS

Cf. A006131, A072183.

Sequence in context: A066552 A206609 A281085 * A214025 A240812 A291425

Adjacent sequences:  A072267 A072268 A072269 * A072271 A072272 A072273

KEYWORD

nonn

AUTHOR

Miklos Kristof, Jul 09 2002

EXTENSIONS

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

STATUS

approved

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Last modified November 12 04:21 EST 2019. Contains 329051 sequences. (Running on oeis4.)