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A250292 Numbers k such that Pell(k) is a semiprime. 0
7, 9, 17, 19, 23, 43, 47, 67, 73, 83, 103, 109, 139, 149, 157, 173, 179, 223, 239, 281, 311, 313, 349, 431, 557, 569, 577, 587 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(29) >= 709. - Hugo Pfoertner, Jul 29 2019

859, 937, 1087, 1151, and 1193 belong to the sequence. 709 and 787 have not yet been ruled out. The next candidate after these appears to be 1471. - Daniel M. Jensen, Oct 18 2019

LINKS

Table of n, a(n) for n=1..28.

FactorDB, Status of Pell(709).

FactorDB, Status of Pell(787).

EXAMPLE

17 is a term since Pell(17) = 1136689 = 137 * 8297 is a semiprime.

MAPLE

pell:= gfun:-rectoproc({a(0)=0, a(1)=1, a(n)=2*a(n-1)+a(n-2)}, a(n), remember):

filter:= proc(n) local F, f;

   F:= ifactors(pell(n), easy)[2];

   if add(f[2], f=F) > 2 then return false fi;

   if hastype(F, symbol) then

     if add(f[2], f=F) >= 2 then return false fi;

   else return evalb(add(f[2], f=F)=2)

   fi;

   F:= ifactors(pell(n))[2];

   evalb(add(f[2], f=F)=2)

end proc:

select(filter, [$1..230]); # Robert Israel, Jan 18 2016

MATHEMATICA

a[0] = 0; a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + a[n - 2]; Select[Range[0, 160], PrimeOmega@ a@ # == 2 &] (* Michael De Vlieger, Jan 19 2016 *)

CROSSREFS

Cf. A001358, A096650, A072381, A085726, A000129.

Sequence in context: A020691 A145830 A195563 * A140364 A022321 A194830

Adjacent sequences:  A250289 A250290 A250291 * A250293 A250294 A250295

KEYWORD

nonn,more,hard

AUTHOR

Eric Chen, Dec 24 2014

EXTENSIONS

a(22)-a(23) from Daniel M. Jensen, Jan 18 2016

a(24) from Arkadiusz Wesolowski, Jan 19 2016

a(25)-a(27) from Sean A. Irvine, Jul 17 2017

a(28) from Sean A. Irvine, Jan 24 2018

STATUS

approved

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Last modified August 7 23:50 EDT 2022. Contains 355995 sequences. (Running on oeis4.)