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

%I A306043 #22 Jul 21 2018 07:53:26
%S A306043 1,4,9,25,49,64,484,625,1225,2209,12100,57600,67600,287296,1517824,
%T A306043 7452900,19492225,64352484,161391616,976375009,3339684100,9758278656,
%U A306043 33371982400,81598207716,448192758784,1641916765129,4148028762241,23794464493849
%N A306043 Lexicographically first sequence of distinct positive squares, no two or more of which sum to a square.
%C A306043 If the squares were not required to be distinct, sequence A305884 would result.
%e A306043 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.
%e A306043 a(4) cannot be 16, because 16 + a(3) = 16 + 9 = 25 = 5^2, but a(4) = 25 satisfies the definition.
%e A306043 a(5) cannot be 36, because 36 + 9 + 4 = 49 = 7^2, but a(5) = 49 satisfies the definition.
%t A306043 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 *)
%o A306043 (Python)
%o A306043 from itertools import combinations
%o A306043 from sympy import integer_nthroot
%o A306043 A306043_list, n, m = [], 1, 1
%o A306043 while len(A306043_list) < 30:
%o A306043     for l in range(1,len(A306043_list)+1):
%o A306043         for d in combinations(A306043_list,l):
%o A306043             if integer_nthroot(sum(d)+m,2)[1]:
%o A306043                 break
%o A306043         else:
%o A306043             continue
%o A306043         break
%o A306043     else:
%o A306043         A306043_list.append(m)
%o A306043     n += 1
%o A306043     m += 2*n-1 # _Chai Wah Wu_, Jun 19 2018
%Y A306043 Cf. A305884.
%K A306043 nonn
%O A306043 1,2
%A A306043 _Jon E. Schoenfield_, Jun 17 2018
%E A306043 a(24)-a(26) from _Giovanni Resta_, Jun 19 2018
%E A306043 a(27)-a(28) from _Jon E. Schoenfield_, Jul 21 2018