A305884 Lexicographically first sequence of positive squares, no two or more of which sum to a square.

1, 1, 1, 4, 16, 25, 25, 324, 841, 1849, 2601, 14884, 18769, 103041, 292681, 774400, 3400336, 13307904, 34892649, 179399236, 582643044, 2008473856, 4369606609, 22833627664, 67113119844, 251608579236, 1240247504896, 3174109249609

Offset: 1

Author

Jon E. Schoenfield, Jun 13 2018

Keywords

• nonn
• more

Comments

Conjecture: the only values that appear more than once are 1 and 25.

If it were required that the terms be distinct, A306043 would result.

Examples

			All terms are positive, so a(1) = 1; likewise, a(2) = a(3) = 1.
a(4) cannot be 1, because the first 4 terms would then sum to 4 = 2^2; however, no two or more terms of {1, 1, 1, 4} sum to a square, so a(4) = 4.
a(5) cannot also be 4, because 4 + 4 + 1 = 9 = 3^2, nor can it be 9, since 9 + 4 + 1 + 1 + 1 = 16 = 4^2, but a(5) = 16 satisfies the definition.
		

Crossrefs

Cf. A000290, A064776, A306043.

Programs

Mathematica

a = {1}; Do[n = Last@a; s = Select[Union[Total /@ Subsets[a^2]], # >= n &];
While[AnyTrue[s, IntegerQ@ Sqrt[n^2 + #] &], n++]; AppendTo[a, n], {14}]; a^2 (* Giovanni Resta, Jun 14 2018 *)

Python

from sympy import integer_nthroot
from sympy.utilities.iterables import multiset_combinations
A305884_list, n, m = [], 1, 1
while len(A305884_list) < 30:
    for l in range(1,len(A305884_list)+1):
        for d in multiset_combinations(A305884_list,l):
            if integer_nthroot(sum(d)+m,2)[1]:
                break
        else:
            continue
        break
    else:
        A305884_list.append(m)
        continue
    n += 1
    m += 2*n-1 # Chai Wah Wu, Jun 19 2018

Extensions

a(25)-a(26) from Giovanni Resta, Jun 14 2018

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