A232176 - OEIS

%I #34 Jan 09 2026 14:24:16

%S 1,2,6,10,14,18,7,5,8,34,6,42,46,15,54,16,14,66,70,74,23,82,9,90,17,

%T 98,102,10,110,15,25,122,126,16,39,48,40,21,150,34,158,29,54,48,30,13,

%U 182,63,55,194,56,202,14,45,214,63,222,26,41,234,31,42,39,250,32,63

%N Least positive k such that n^2 + triangular(k) is a square.

%C Triangular(k) = A000217(k) = k*(k+1)/2.

%C a(n) <= 4*n - 2, because with k = 4*n-2: n^2 + k*(k+1)/2 = n^2 + (4*n-2)*(4*n-1)/2 = 9*n^2 - 6*n + 1 = (3*n-1)^2.

%C The sequence of numbers n such that a(n)=n begins: 8, 800, 7683200 ... - a subsequence of A220186.

%H Andrey Zabolotskiy, <a href="/A232176/b232176.txt">Table of n, a(n) for n = 0..5000</a>

%t lpk[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[n^2+(k(k+1))/2]],k++];k]; Array[ lpk,70,0] (* _Harvey P. Dale_, May 04 2018 *)

%o (Python)

%o import math

%o def a(n):

%o   for k in range(1,max(1,4*n-2)+1):

%o     sum = n*n + k*(k+1)//2

%o     r = math.isqrt(sum)

%o     if r*r == sum:

%o       return k

%o print([a(n) for n in range(70)])

%o (PARI) a(n) = {k = 1; while (! issquare(n^2 + k*(k+1)/2), k++); k;} \\ _Michel Marcus_, Nov 20 2013

%Y Cf. A000290, A000217, A038202, A055527, A220186, A232175.

%Y Cf. A232179 (least k>=0 such that n^2 + triangular(k) is a triangular number).

%Y Cf. A101157 (least k>0 such that triangular(n) + k^2 is a triangular number).

%Y Cf. A232178 (least k>=0 such that triangular(n) + k^2 is a square).

%K nonn

%O 0,2

%A _Alex Ratushnyak_, Nov 19 2013