A155707 Numbers expressible as a^2 + k b^2 with nonzero integers a,b, for k=2, k=3, k=5 and k=7.

144, 576, 1009, 1129, 1201, 1296, 1801, 1849, 2304, 2521, 2689, 2881, 3049, 3361, 3529, 3600, 3889, 4036, 4201, 4356, 4489, 4516, 4561, 4729, 4804, 5184, 5209, 5569, 5881, 5929, 6841, 7009, 7056, 7204, 7396, 7561, 7681, 8089, 8521, 8689, 8761, 8929

Offset: 1

Author

M. F. Hasler, Feb 10 2009

Keywords

• nonn

Comments

Subsequence of A155708.

Links

• Robert Israel, Table of n, a(n) for n = 1..2500

Crossrefs

Subsequence of A155708.

Cf. A028372, A000404, A154777, A092572, A097268, A155714, A155560.

Programs

Maple

filter:= proc(x) local k,S;
   if numtheory:-quadres(x,3*5*7)<> 1 then return false fi;
   for k in [2,3,5,7] do
     S:= [isolve(x = a^2 + k*b^2)];
     if andmap(t -> subs(t,a*b) = 0, S) then return false fi;
   od;
   true
end proc;
select(filter, [$1..10000]); # Robert Israel, May 14 2025

PARI

isA155707(n,/* optional 2nd arg allows us to get other sequences */c=[7, 5, 3, 2]) = { for(i=1, #c, for(b=1, sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return); 1}
for(n=1,9999, isA155707(n) & print1(n","))