login
A250292
Numbers k such that Pell(k) is a semiprime.
1
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
1,1
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
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
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