OFFSET
1,4
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(Python)
from sympy import integer_nthroot
from sympy.utilities.iterables import multiset_combinations
A305884_list, n, m = [], 1, 1
while len(A305884_list) < 30:
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:
A305884_list.append(m)
continue
n += 1
m += 2*n-1 # Chai Wah Wu, Jun 19 2018
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jon E. Schoenfield, Jun 13 2018
EXTENSIONS
a(25)-a(26) from Giovanni Resta, Jun 14 2018
a(27)-a(28) from Jon E. Schoenfield, Jul 21 2018
STATUS
approved