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A377084 a(n) is the maximum integer for which some minimum-length sum equaling a(n) of perfect squares less than n^2 excludes (n-1)^2.

%I A377084 #12 Oct 16 2024 20:53:54
%S A377084 3,16,31,128,191,324,368,561,891,1200,1104,1993,2535,2692,2896,4321,
%T A377084 4880,5832,5776,7485,9144,9680,8903,14353,15576,14872,14527,22736,
%U A377084 21697,22500,22587,30537,31451,33524,30076,42768,43664,43097,39317,59200,58835,59893
%N A377084 a(n) is the maximum integer for which some minimum-length sum equaling a(n) of perfect squares less than n^2 excludes (n-1)^2.
%H A377084 Nathan Fox, <a href="/A377084/b377084.txt">Table of n, a(n) for n = 3..100</a>
%H A377084 N. Bradley Fox et al., <a href="https://arxiv.org/abs/2409.09863">Elated Numbers</a>, arXiv:2409.09863 [math.NT], 2024.
%e A377084 561 cannot be written as a sum of fewer than nine perfect squares less than 10^2. 561 can be written as a sum of nine of these numbers in five ways:
%e A377084 561 = 1^2 + 5^2 + 7^2 + 9^2 + 9^2 + 9^2 + 9^2 + 9^2 + 9^2
%e A377084 561 = 3^2 + 6^2 + 8^2 + 8^2 + 8^2 + 9^2 + 9^2 + 9^2 + 9^2
%e A377084 561 = 3^2 + 7^2 + 7^2 + 7^2 + 9^2 + 9^2 + 9^2 + 9^2 + 9^2
%e A377084 561 = 5^2 + 5^2 + 5^2 + 9^2 + 9^2 + 9^2 + 9^2 + 9^2 + 9^2
%e A377084 561 = 7^2 + 8^2 + 8^2 + 8^2 + 8^2 + 8^2 + 8^2 + 8^2 + 8^2
%e A377084 The last sum here does not include 9^2, so a(10) >= 561. In fact, a(10) = 561, as every number larger than 561 has 9^2 in every shortest decomposition of this form.
%Y A377084 Cf. A377085.
%K A377084 nonn
%O A377084 3,1
%A A377084 N. Bradley Fox, _Nathan Fox_, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024