cp's OEIS Frontend

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

%I A305884 #21 Jul 22 2018 06:54:58
%S A305884 1,1,1,4,16,25,25,324,841,1849,2601,14884,18769,103041,292681,774400,
%T A305884 3400336,13307904,34892649,179399236,582643044,2008473856,4369606609,
%U A305884 22833627664,67113119844,251608579236,1240247504896,3174109249609
%N A305884 Lexicographically first sequence of positive squares, no two or more of which sum to a square.
%C A305884 Conjecture: the only values that appear more than once are 1 and 25.
%C A305884 If it were required that the terms be distinct, A306043 would result.
%e A305884 All terms are positive, so a(1) = 1; likewise, a(2) = a(3) = 1.
%e A305884 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.
%e A305884 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.
%t A305884 a = {1}; Do[n = Last@a; s = Select[Union[Total /@ Subsets[a^2]], # >= n &];
%t A305884 While[AnyTrue[s, IntegerQ@ Sqrt[n^2 + #] &], n++]; AppendTo[a, n], {14}]; a^2 (* _Giovanni Resta_, Jun 14 2018 *)
%o A305884 (Python)
%o A305884 from sympy import integer_nthroot
%o A305884 from sympy.utilities.iterables import multiset_combinations
%o A305884 A305884_list, n, m = [], 1, 1
%o A305884 while len(A305884_list) < 30:
%o A305884     for l in range(1,len(A305884_list)+1):
%o A305884         for d in multiset_combinations(A305884_list,l):
%o A305884             if integer_nthroot(sum(d)+m,2)[1]:
%o A305884                 break
%o A305884         else:
%o A305884             continue
%o A305884         break
%o A305884     else:
%o A305884         A305884_list.append(m)
%o A305884         continue
%o A305884     n += 1
%o A305884     m += 2*n-1 # _Chai Wah Wu_, Jun 19 2018
%Y A305884 Cf. A000290, A064776, A306043.
%K A305884 nonn,more
%O A305884 1,4
%A A305884 _Jon E. Schoenfield_, Jun 13 2018
%E A305884 a(25)-a(26) from _Giovanni Resta_, Jun 14 2018
%E A305884 a(27)-a(28) from _Jon E. Schoenfield_, Jul 21 2018