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A305884
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Lexicographically first sequence of positive squares, no two or more of which sum to a square.
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1
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(Python)
from sympy import integer_nthroot
from sympy.utilities.iterables import multiset_combinations
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:
continue
n += 1
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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