A105606 -id:A105606 - cp's OEIS frontend

A008555 Primitive parts of Pell numbers.

1, 2, 5, 6, 29, 7, 169, 34, 197, 41, 5741, 33, 33461, 239, 1345, 1154, 1136689, 199, 6625109, 1121, 45697, 8119, 225058681, 1153, 45232349, 47321, 7761797, 38081, 44560482149, 961, 259717522849, 1331714, 52734529, 1607521, 1800193921, 39201

Offset: 1

N. J. A. Sloane

nonn

Also called Sylvester-Pell cyclotomic numbers. - Paul Barry, Apr 15 2005

According to Guy, Raphael Robinson noticed that a(7) and a(30) are squares and asked if there are more. There are no others in the first 10000 terms. [T. D. Noe, May 07 2009]

			a(8)=34 because pell(8)=408 and 408/(a(4)*a(2)*a(1)) = 408/12 = 34. [From _T. D. Noe_, May 07 2009]

R. K. Guy, Unsolved Problems in Number Theory, A3.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number. - _Paul Barry_, Apr 15 2005

Cf. A061446 (primitive part of Fibonacci numbers). [T. D. Noe, May 07 2009]

Cf. A105606.

pell={1,2}; pp={1,2}; Do[s=2*pell[[ -1]]+pell[[ -2]]; AppendTo[pell,s]; AppendTo[pp, s/Times@@pp[[Most[Divisors[n]]]]], {n,3,40}]; pp (* T. D. Noe, May 07 2009 *)

a(n) = A000129(n) / Product_{dT. D. Noe, May 07 2009]

a(n) = Product_{k=1..n-1} if(gcd(n, k)=1, (1+sqrt(2))-(1-sqrt(2))*exp(2*Pi*I*k/n), 1), I=sqrt(-1). - Paul Barry, Apr 15 2005

Corrected and extended by T. D. Noe, May 07 2009

Edited by N. J. A. Sloane, Oct 04 2009

A105607 Sylvester cyclotomic numbers for A002605.

Original entry on oeis.org

1, 2, 6, 8, 44, 10, 328, 56, 408, 76, 18272, 52, 136384, 568, 3856, 3104, 7598336, 424, 56714752, 2896, 215104, 31648, 3159738368, 3088, 536013824, 236224, 71910912, 161344, 1313964867584, 2320, 9807567290368, 9634304, 667730944, 13160704, 37860806656, 172864, 4078438577864704, 98232832, 37201186816, 9584896

Offset: 1

Paul Barry, Apr 15 2005

nonn

Primitive parts of A002605.

Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.

Cf. A002605, A105608, A105609.

f[n_] := FullSimplify[ Expand[ Times @@ ((1+Sqrt[3])-(1-Sqrt[3])*Exp[2Pi*I*Select[Range[n-1], GCD[ #, n] == 1 &]/n])]]; Table[ f[n], {n, 1, 32}] (* Robert G. Wilson v, Aug 02 2005 *)

a(n) = A002605(n)/A105606(n); a(n) = Product_{k=1..n-1, gcd(n, k)=1} (1+sqrt(3))-(1-sqrt(3))*exp(2*Pi*i*k/n), i = sqrt(-1). - Robert G. Wilson v, Aug 02 2005

More terms from David Wasserman, May 06 2008

cp's OEIS Frontend

A008555 Primitive parts of Pell numbers.

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