A230044 - OEIS

%I #36 Mar 27 2021 23:33:26

%S 0,1,2,3,5,6,9,10,11,12,14,15,17,19,20,21,24,27,28,29,30,32,35,36,39,

%T 41,42,44,45,46,50,51,53,54,55,56,57,62,65,66,69,71,72,74,75,77,78,80,

%U 82,84,87,89,90,91,95,96,100,101,104,105,107,109,110,111,116,117,119,120,122,126,127,128

%N Nonnegative numbers k such that k plus a perfect square is a triangular number.

%C Negative k are in A175035.

%C Numbers such that the Diophantine equation y^2 + y - 2x^2 = 2n, y > 0 has a solution. Empirically, solutions (x,y) don't exceed (5n,5n) for n < 10^5. Record quotients y/n are at n = 2, 3, 12, 45, 1225, 6806, ...

%C Conjecture: these are the sorted distinct terms of A064784.

%C n is in this sequence iff 8n+1 is in A035251, that is, every prime p == 3 or 5 (mod 8) dividing 8n+1 appears to an even power. - _Max Alekseyev_, Oct 14 2013

%H Charles R Greathouse IV, <a href="/A230044/b230044.txt">Table of n, a(n) for n = 1..10000</a>

%e 28 is triangular, and 25 is a square <= 28, and 28-25=3, so 3 is in sequence.

%o (PARI) B=bnfinit(z^2-8); is(n)=#bnfisintnorm(B,8*n+1) \\ _Max Alekseyev_, Oct 13 2013

%Y Cf. A035251, A064784, A175035.

%K nonn

%O 1,3

%A _Ralf Stephan_, Oct 06 2013