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Primitive parts of Pell numbers.
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%I #17 Mar 14 2023 03:40:19

%S 1,2,5,6,29,7,169,34,197,41,5741,33,33461,239,1345,1154,1136689,199,

%T 6625109,1121,45697,8119,225058681,1153,45232349,47321,7761797,38081,

%U 44560482149,961,259717522849,1331714,52734529,1607521,1800193921,39201

%N Primitive parts of Pell numbers.

%C Also called Sylvester-Pell cyclotomic numbers. - _Paul Barry_, Apr 15 2005

%C 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]

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

%H T. D. Noe, <a href="/A008555/b008555.txt">Table of n, a(n) for n=1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SylvesterCyclotomicNumber.html">Sylvester Cyclotomic Number</a>. - _Paul Barry_, Apr 15 2005

%F a(n) = A000129(n) / Product_{d<n,d|n} a(d). [_T. D. Noe_, May 07 2009]

%F 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

%e 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]

%t 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 *)

%Y Cf. A061446 (primitive part of Fibonacci numbers). [_T. D. Noe_, May 07 2009]

%Y Cf. A105606.

%K nonn

%O 1,2

%A _N. J. A. Sloane_.

%E Corrected and extended by _T. D. Noe_, May 07 2009

%E Edited by _N. J. A. Sloane_, Oct 04 2009