A306043 Lexicographically first sequence of distinct positive squares, no two or more of which sum to a square.

1, 4, 9, 25, 49, 64, 484, 625, 1225, 2209, 12100, 57600, 67600, 287296, 1517824, 7452900, 19492225, 64352484, 161391616, 976375009, 3339684100, 9758278656, 33371982400, 81598207716, 448192758784, 1641916765129, 4148028762241, 23794464493849

Offset: 1

Author

Jon E. Schoenfield, Jun 17 2018

Keywords

• nonn

Comments

If the squares were not required to be distinct, sequence A305884 would result.

Examples

			All terms are distinct positive squares, and no two or more of the first three positive squares sum to a square, so a(1) = 1^2 = 1, a(2) = 2^2 = 4, and a(3) = 3^2 = 9.
a(4) cannot be 16, because 16 + a(3) = 16 + 9 = 25 = 5^2, but a(4) = 25 satisfies the definition.
a(5) cannot be 36, because 36 + 9 + 4 = 49 = 7^2, but a(5) = 49 satisfies the definition.
		

Crossrefs

Cf. A305884.

Programs

Mathematica

a = {1}; Do[n = 1 + Last@a; s = Select[Union[Total /@ Subsets[a^2]], # >= n &]; While[AnyTrue[s, IntegerQ@Sqrt[n^2 + #] &], n++]; AppendTo[a, n], {12}]; a^2 (* Giovanni Resta, Jun 19 2018 *)

Python

from itertools import combinations
from sympy import integer_nthroot
A306043_list, n, m = [], 1, 1
while len(A306043_list) < 30:
    for l in range(1,len(A306043_list)+1):
        for d in combinations(A306043_list,l):
            if integer_nthroot(sum(d)+m,2)[1]:
                break
        else:
            continue
        break
    else:
        A306043_list.append(m)
    n += 1
    m += 2*n-1 # Chai Wah Wu, Jun 19 2018

Extensions

a(24)-a(26) from Giovanni Resta, Jun 19 2018

a(27)-a(28) from Jon E. Schoenfield, Jul 21 2018