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
Keywords
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
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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
Comments