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

1,2

Comments

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

Links

Table of n, a(n) for n=1..28.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A305884.

Sequence in context: A069557 A230312 A332646 * A194269 A336230 A130283

Adjacent sequences: A306040 A306041 A306042 * A306044 A306045 A306046

Keyword

nonn

Author

Jon E. Schoenfield, Jun 17 2018

Extensions

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

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

Status

approved