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A381811 The largest nonnegative integer j for which each integer n,n+2,...,n+2j can be written as the sum of the squares for some partition of n.

%I A381811 #15 May 12 2025 00:18:12
%S A381811 0,1,1,3,4,4,7,13,13,18,25,25,32,32,40,49,52,62,73,85,102,112,127,133,
%T A381811 160,166,166,184,203,208,228,249,271,294,322,343,373,376,376,403,431,
%U A381811 490,521,521,553,592,620,655,662,662,735,735,773,812,852,893,901,943,986
%N A381811 The largest nonnegative integer j for which each integer n,n+2,...,n+2j can be written as the sum of the squares for some partition of n.
%C A381811 a(n) has an asymptotic equivalence with (1/2)*n^2-sqrt(2)*n^(3/2)+O(n^(5/4)) (Reznick 1989, p. 201).
%H A381811 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 A381811 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 A381811 a(n) = (A383682(n) - n) / 2.
%e A381811 a(3) = 1, because n, n+2 (3 and 5) can be written as the sum of the squares for some partition of n; 3=1^2+1^2+1^2 and 5=2^2+1^2. However, 7 cannot be written as the sum of squares of the parts of a partition of 3, so a(3) = 1.
%e A381811 a(4) = 3, because n, n+2, n+4 and n+6 (4, 6, 8 and 10) can be written as the sum of the squares for some partition of n; 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. However, 12 cannot be written as the sum of squares of the parts of a partition of 4, so a(4) = 3.
%Y A381811 Cf. A069999 (a(n) provides a lower bound for A069999(n)).
%Y A381811 Cf. A000041, A383682.
%K A381811 nonn
%O A381811 1,4
%A A381811 _Noah A Rosenberg_, May 05 2025