A328343 Numbers k such that it is possible to find k consecutive squares whose sum is equal to the sum of two consecutive squares.

1, 2, 3, 10, 17, 25, 26, 34, 41, 50, 51, 65, 73, 82, 89, 97, 106, 113, 122, 123, 145, 146, 169, 170, 178, 185, 194, 218, 219, 241, 250, 257, 267, 274, 281, 291, 298, 305, 314, 338, 339, 353, 362, 370, 377, 386, 394, 401, 409, 410, 411, 433, 449, 457, 505, 530, 545

Offset

1,2

Comments

The program generates solutions to the problem, but not necessarily all solutions. To exclude the existence of further solution you have to apply coset arithmetics (modulo operations).

Links

Table of n, a(n) for n=1..57.

Example

k = 1: 3^2 + 4^2 = 5^2.
k = 2: 3^2 + 4^2 = 3^2 + 4^2.
k = 3: 13^2 + 14^2 = 10^2 + 11^2 + 12^2.
k = 10: 26^2 + 27^2 = 7^2 + ... + 16^2.
k = 17: 40^2 + 41^2 = 5^2 + ... + 21^2.
k = 25: 78^2 + 79^2 = 9^2 + ... + 33^2.
k = 26: 205^2 + 206^2 = 44^2 + ... + 49^2.
k = 34: 856^2 + 857^2 = 191^2 + ... + 224^2.
k = 41: 3029^2 + 3030^2 = 649^2 + ... + 689^2.
k = 50: 146^2 + 147^2 = 1^2 + ... + 50^2.
k = 51: 210^2 + 211^2 = 14^2 + ... + 64^2.
k = 65: 236^2 + 237^2 = 5^2 + ... + 69^2.
k = 73: 278^2 + 279^2 = 5^2 + ... + 76^2.
k = 82: 1070^2 + 1071^2 = 125^2 + ... + 206^2.
k = 89: 147445^2 + 147446^2 = 22059^2 + ... + 22147^2.
k = 97: 544^2 + 545^2 = 25^2 + ... + 121^2.

Mathematica

Select[Range[60], {} != FindInstance[ Sum[t^2, {t, x, x+#-1}] == y^2 + (y + 1)^2, {x, y}, Integers] &] (* Giovanni Resta, Oct 23 2019 *)

Prog

(Python)
import math
for n in range(1, 100):
   for b in range(1, 10000000):
      d = (6*b*b*(n+1)+6*b*n*(n+1)+2*n*n*n+3*n*n+n)
      w = int((math.sqrt(d/6)))
      a = w
      if 6*a*a-6*b*b*(n+1)-6*b*n*(n+1)-2*n*n*n-3*n*n-n == 0:
         print(a, b, n+1)
      a = w+1
      if 6*a*a-6*b*b*(n+1)-6*b*n*(n+1)-2*n*n*n-3*n*n-n == 0:
         print(a, b, n+1)
      a = w-1
      if 6*a*a-6*b*b*(n+1)-6*b*n*(n+1)-2*n*n*n-3*n*n-n == 0:
         print(a, b, n+1)

Crossrefs

Cf. A001032, A185545.

Sequence in context: A357271 A060744 A213391 * A309350 A300285 A192798

Adjacent sequences: A328340 A328341 A328342 * A328344 A328345 A328346

Keyword

nonn

Author

Reiner Moewald, Oct 13 2019

Extensions

a(17)-a(38) from Jon E. Schoenfield, Oct 22 2019

a(39)-a(57) from Jon E. Schoenfield and Giovanni Resta, Oct 23 2019

Status

approved