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A306043
Lexicographically first sequence of distinct positive squares, no two or more of which sum to a square.
1
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.
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
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