A390086 - OEIS

%I #48 Dec 09 2025 16:26:49

%S 0,1,3,4,6,8,11,14,18,21,25,30,34,39,45,50

%N a(n) is the smallest index k such that A390084(k) = n.

%C In the graph of A390084, 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 n, there are infinitely many such sets, and at least one of them attains the smallest scale — this minimal scale is denoted as a(n).

%C If n = 3*k, let m = n*(n+3)/6, then {m..m-k+1, k-m..2*k-m; k-1..1} is an irreducible zero-sum integer set of length n, another is {m..m-k+2, k-1-m..2*k-2-m, k+1..1}. so, a(n) <= m = n*(n+3)/6.

%C If n = 3*k+1, let m = (n-1)*(n+4)/6, then {m..m-k+1, k-m..2*k-m, k..1} is an irreducible zero-sum integer set of length n. so, a(n) <= m = (n-1)*(n+4)/6 < n*(n+3)/6.

%C If n = 3*k+2, let m = (n-1)*(n+4)/6, then {m..m-k+1, k-m..2*k-m, k+1..1} is an irreducible zero-sum integer set of length n. so, a(n) <= m = (n-1)*(n+4)/6 < n*(n+3)/6.

%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+3)/6.

%e For n = 1, a(1) = 0, since the longest irreducible zero-sum integer set with length 1 is {0}.

%e For n = 6, a(6) = 8, since the irreducible zero-sum integer set with length 6 and min-scale is +-{8, -7, 6, -5, -4, 2}. There are infinitely many such sets with length 6, e.g. +-{9, -8, -7, 3, 2, 1}, but its scale 9 > 8.

%e From _Bert Dobbelaere_, Nov 09 2025: (Start)

%e the irreducible zero-sum integer set with min-scale:

%e   a(1) = 0: {0}

%e   a(2) = 1: {1, -1}

%e   a(3) = 3: +-{3, -2, -1}

%e   a(4) = 4: +-{4, -3, -2, 1}

%e   a(5) = 6: +-{6, -5, -4, 2, 1}

%e   a(6) = 8: +-{8, -7, 6, -5, -4, 2}

%e   a(7) = 11: +-{11, 10, -9, -8, -7, 2, 1}

%e   a(8) = 14: +-{14, 13, -12, -11, -10, 3, 2, 1}

%e   a(9) = 18: +-{18, 17, 16, -15, -14, -13, -12, 2, 1}

%e   a(10) = 21: +-{21, 20, 19, -18, -17, -16, -15, 3, 2, 1}

%e   a(11) = 25: +-{25, 24, 23, -22, -21, -20, -19, 4, 3, 2, 1}

%e   a(12) = 30: +-{30, 29, 28, 27, -26, -25, -24, -23, -22, 3, 2, 1}

%e   a(13) = 34: +-{34, 33, 32, 31, -30, -29, -28, -27, -26, 4, 3, 2, 1}

%e   a(14) = 39: +-{39, 38, 37, 36, -35, -34, -33, -32, -31, 5, 4, 3, 2, 1}

%e   a(15) = 45: +-{45, 44, 43, 42, 41, -40, -39, -38, -37, -36, -35, 4, 3, 2, 1}

%e   a(16) = 50: +-{50, 49, 48, 47, 46, -45, -44, -43, -42, -41, -40, 5, 4, 3, 2, 1}

%e (End)

%Y Cf. A390084, A301654.

%K nonn,more

%O 1,3

%A _Hu Junhua_, Nov 13 2025