cp's OEIS Frontend

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

%I A362528 #4 Apr 23 2023 22:25:48
%S A362528 11,27,488,683,852,907,964,1372,1445,3971,5947,6563,8587,40003,70803,
%T A362528 111603,116285,129603,133958,291607,465125,1229884,1555208,2088027,
%U A362528 37442165,89629867,93896107,149768645,197712043,287946964,298391123
%N A362528 Numbers that can be written in at least 3 ways as the sum of a Lucas number (A000032) and a square.
%C A362528 Numbers k such that k = A000032(x) + y^2 for x, y >= 0 has at least 3 solutions.
%C A362528 Conjecture: there are never more than 3 solutions.
%e A362528 a(1) = 11 = A000032(0) + 3^2 = A000032(4) + 2^2 = A000032(5) + 0^2.
%e A362528 a(2) = 27 = A000032(0) + 5^2 = A000032(5) + 4^2 = A000032(6) + 3^2.
%e A362528 a(3) = 488 = A000032(3) + 22^2 = A000032(8) + 21^2 = A000032(11) + 17^2.
%p A362528 N:= 3*10^8: # for terms <= N
%p A362528 luc:= n -> combinat:-fibonacci(n-1) + combinat:-fibonacci(n+1):
%p A362528 S:= {}:
%p A362528 for x from 1 to floor(sqrt(N)) do
%p A362528   s:= x^2;
%p A362528   for i from 2 do
%p A362528     l:= luc(i);
%p A362528     if s+l > N then break fi;
%p A362528     v:= f(s+l);
%p A362528     if v >= 3 and not member(s+l,S) then S:= S union {s+l}; fi
%p A362528 od od:
%p A362528 sort(convert(S,list));
%Y A362528 Cf. A000032, A362434.
%K A362528 nonn
%O A362528 1,1
%A A362528 _Robert Israel_, Apr 23 2023