A362528 Numbers that can be written in at least 3 ways as the sum of a Lucas number (A000032) and a square.

11, 27, 488, 683, 852, 907, 964, 1372, 1445, 3971, 5947, 6563, 8587, 40003, 70803, 111603, 116285, 129603, 133958, 291607, 465125, 1229884, 1555208, 2088027, 37442165, 89629867, 93896107, 149768645, 197712043, 287946964, 298391123

Offset: 1

Author

Robert Israel, Apr 23 2023

Keywords

• nonn

Comments

Numbers k such that k = A000032(x) + y^2 for x, y >= 0 has at least 3 solutions.

Conjecture: there are never more than 3 solutions.

Examples

			a(1) = 11 = A000032(0) + 3^2 = A000032(4) + 2^2 = A000032(5) + 0^2.
a(2) = 27 = A000032(0) + 5^2 = A000032(5) + 4^2 = A000032(6) + 3^2.
a(3) = 488 = A000032(3) + 22^2 = A000032(8) + 21^2 = A000032(11) + 17^2.
		

Crossrefs

Cf. A000032, A362434.

Programs

Maple

N:= 3*10^8: # for terms <= N
luc:= n -> combinat:-fibonacci(n-1) + combinat:-fibonacci(n+1):
S:= {}:
for x from 1 to floor(sqrt(N)) do
  s:= x^2;
  for i from 2 do
    l:= luc(i);
    if s+l > N then break fi;
    v:= f(s+l);
    if v >= 3 and not member(s+l,S) then S:= S union {s+l}; fi
od od:
sort(convert(S,list));