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

1,2

Comments

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]

References

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

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Xuejun Guo and Zhengyu Tao, Counterexamples to Stanley's conjecture on dimer coverings, arXiv:2605.28195 [math.CO], 2026.

Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number. - Paul Barry, Apr 15 2005

Formula

a(n) = A000129(n) / Product_{d<n,d|n} a(d). [T. 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

Example

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]

Mathematica

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 *)

Crossrefs

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

Cf. A105606.

Sequence in context: A214200 A382360 A273924 * A056441 A365085 A164805

Adjacent sequences: A008552 A008553 A008554 * A008556 A008557 A008558

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

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

Status

approved