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%I #30 Dec 02 2025 20:39:57
%S 1,4,6,11,16,23,30
%N a(n) is the smallest index k such that A390085(k) = n.
%C In the graph of A390085, for a given positive integer n, there exists a plateau with height n; a(n) denotes the abscissa corresponding to the left endpoint of this plateau.
%C A zero-sum integer set is called irreducible if it does not contain any smaller zero-sum subset. The maximum absolute value of elements in such a set is herein referred to as its "scale". For a given length 2*n, there are infinitely many irreducible zero-sum odd sets, and at least one of them attains the smallest scale — the index of this minimal scale in positive odd numbers is denoted as a(n).
%C If n = 2*k, let m = n*(n+2)-3, then {m, m-2, ..., m-2*n+6; 2*n-4-m, m-2*n+2; 2*n-m, 2*n+2-m, ..., 4*n-4-m; 4*n-m} is an irreducible zero-sum integer set of length n. so, a(n) <= (m+1)/2 = n*(n+2)/2-1.
%C If n = 2*k+1, let m = n*(n+2)-4, then {m, m-2, ..., m-2*n+6; 2*n-4-m, m-2*n+2; 2*n-m, 2*n+2-m, ..., 4*n-2-m} is an irreducible zero-sum integer set of length n. so, a(n) <= (m+1)/2 = (n-1)*(n+3)/2.
%H Hu Junhua, <a href="https://bbs.emath.ac.cn/thread-2320-1-1.html">Balanced samples of the initial segment of natural numbers</a>, Chinese Math R & D BBS, Apr 21 2010.
%F a(n) <= n*(n+2)/2-1.
%e For n = 1, a(1) = 1, since the irreducible zero-sum odd set with length 2*1 and min-scale is {1, -1}.
%e For n = 2, a(2) = 4, since the irreducible zero-sum odd set with length 2*2 and min-scale is +-{7, -5, -3, 1}, where 7 (4th positive odd number) gives index 4. There are infinitely many irreducible zero-sum odd sets with length 2*2, e.g. +-{9, -7, -5, 3}, but its scale 9 > 7.
%e The irreducible zero-sum odd set with min-scale:
%e a(1) = 1: {1,-1}
%e a(2) = 4: +-{7, -5, -3, 1}
%e a(3) = 6: +-{11, -9, 7, -5, -3, -1}
%e a(4) = 11: +-{21, -19, 17, -15, 11, -7, -5, -3} (from _Christian Sievers_, Nov 12 2025)
%e +-{21, -19, 17, -15, -13, 5, 3, 1}
%e +-{21, 19, -17, 15, -13, -11, -9, -5}
%e a(5) = 16: +-{31, 29, 27, -25, 23, -21, -19, -17, -15, -13}
%e a(6) = 23: +-{45, 43, 41, 39, -37, 35, -33, -31, -29, -27, -25, -21}
%e a(7) = 30: +-{59, 57, 55, 53, 51, -49, 47, -45, -43, -41, -39, -37, -35, -33}
%Y Cf. A390085, A116940.
%K nonn,more
%O 1,2
%A _Hu Junhua_, Nov 13 2025