A250292 Numbers k such that Pell(k) is a semiprime.

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

Offset: 1

Author

Eric Chen, Dec 24 2014

Keywords

• nonn
• more
• hard

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

Examples

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

Links

• FactorDB, Status of Pell(709).
• FactorDB, Status of Pell(787).

Crossrefs

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

Programs

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 *)

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