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A383682 The largest nonnegative integer value of j for which each integer n, n+2, ..., j-4, j-2, j can be written as the sum of the squares of the elements of a partition of n.

%I A383682 #9 May 10 2025 19:31:33
%S A383682 1,4,5,10,13,14,21,34,35,46,61,62,77,78,95,114,121,142,165,190,225,
%T A383682 246,277,290,345,358,359,396,435,446,487,530,575,622,679,722,783,790,
%U A383682 791,846,903,1022,1085,1086,1151,1230,1287,1358,1373,1374,1521,1522,1599
%N A383682 The largest nonnegative integer value of j for which each integer n, n+2, ..., j-4, j-2, j can be written as the sum of the squares of the elements of a partition of n.
%H A383682 B. Reznick, <a href="https://doi.org/10.1016/0022-314X(89)90006-1">The sum of the squares of the parts of a partition, and some related questions</a>, J. Number Theory 33 (1989), 199-208.
%H A383682 P. Winkler, <a href="https://doi.org/10.1016/0166-218X(90)90137-2">Mean distance in a tree</a>, Discr. Appl. Math. (1990), 179-185.
%F A383682 a(n) ~ n^2-2*sqrt(2)*n^(3/2)+O(n^(5/4)) (Reznick 1989, p. 201).
%e A383682 Consider n=3: 3 and 5 can be written as sums of squares of partitions of 3, as 3=1^2+1^2+1^2 and 5=2^2+1^2, but 7 cannot be written as a sum of squares of a partition of 3, so a(3)=5.
%e A383682 Consider n=4: 4, 6, 8, and 10 can be written as sums of squares of partitions of 4, as 4=1^2+1^2+1^2+1^2, 6=2^2+1^2+1^2, 8=2^2+2^2, and 10=3^2+1^2, but 12 cannot be written as a sum of squares of a partition of 4, so a(4)=10.
%Y A383682 Cf. A381811.
%K A383682 nonn
%O A383682 1,2
%A A383682 _Noah A Rosenberg_, May 05 2025