A232175 -id:A232175 - cp's OEIS frontend

A232176 Least positive k such that n^2 + triangular(k) is a square.

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, 98, 102, 10, 110, 15, 25, 122, 126, 16, 39, 48, 40, 21, 150, 34, 158, 29, 54, 48, 30, 13, 182, 63, 55, 194, 56, 202, 14, 45, 214, 63, 222, 26, 41, 234, 31, 42, 39, 250, 32, 63

Offset: 0

Alex Ratushnyak, Nov 19 2013

nonn

Triangular(k) = A000217(k) = k*(k+1)/2.
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.
The sequence of numbers n such that a(n)=n begins: 8, 800, 7683200 ... - a subsequence of A220186.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..5000

Cf. A000290, A000217, A038202, A055527, A220186, A232175.
Cf. A232179 (least k>=0 such that n^2 + triangular(k) is a triangular number).
Cf. A101157 (least k>0 such that triangular(n) + k^2 is a triangular number).
Cf. A232178 (least k>=0 such that triangular(n) + k^2 is a square).

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 *)

a(n) = {k = 1; while (! issquare(n^2 + k*(k+1)/2), k++); k;} \\ Michel Marcus, Nov 20 2013

import math
for n in range(77):
  n2 = n*n
  y=1
  for k in range(1,10000001):
    sum = n2 + k*(k+1)//2
    r = int(math.sqrt(sum))
    if r*r == sum:
      print(str(k), end=',')
      y=0
      break
  if y: print('-', end=',')

A232267 Least k>=0 such that n^3 + k^2 is a square.

Original entry on oeis.org

0, 0, 1, 3, 0, 10, 3, 21, 8, 0, 15, 55, 6, 78, 35, 15, 0, 136, 27, 171, 10, 42, 99, 253, 10, 0, 143, 81, 42, 406, 15, 465, 64, 88, 255, 35, 0, 666, 323, 91, 3, 820, 21, 903, 55, 66, 483, 1081, 48, 0, 125, 85, 39, 1378, 81, 165, 28, 76, 783, 1711, 15, 1830, 899, 63

Offset: 0

Alex Ratushnyak, Nov 21 2013

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A000290, A000578, A232175.

Table[k = 0; While[! IntegerQ[Sqrt[n^3 + k^2]], k++]; k, {n, 0, 100}] (* T. D. Noe, Nov 22 2013 *)

cp's OEIS Frontend

A232176 Least positive k such that n^2 + triangular(k) is a square.

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