A250292 - OEIS

%I #62 Dec 01 2019 10:57:28

%S 7,9,17,19,23,43,47,67,73,83,103,109,139,149,157,173,179,223,239,281,

%T 311,313,349,431,557,569,577,587

%N Numbers k such that Pell(k) is a semiprime.

%C a(29) >= 709. - _Hugo Pfoertner_, Jul 29 2019

%C 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

%H FactorDB, <a href="http://factordb.com/index.php?id=1100000000707746051">Status of Pell(709)</a>.

%H FactorDB, <a href="http://factordb.com/index.php?id=1100000000712605734">Status of Pell(787)</a>.

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

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

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

%p    F:= ifactors(pell(n),easy)[2];

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

%p    if hastype(F,symbol) then

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

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

%p    fi;

%p    F:= ifactors(pell(n))[2];

%p    evalb(add(f[2],f=F)=2)

%p end proc:

%p select(filter, [$1..230]); # _Robert Israel_, Jan 18 2016

%t 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 *)

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

%K nonn,more,hard

%O 1,1

%A _Eric Chen_, Dec 24 2014

%E a(22)-a(23) from _Daniel M. Jensen_, Jan 18 2016

%E a(24) from _Arkadiusz Wesolowski_, Jan 19 2016

%E a(25)-a(27) from _Sean A. Irvine_, Jul 17 2017

%E a(28) from _Sean A. Irvine_, Jan 24 2018