The fourth positive element in the greedy $B_h$-set

Abstract

For $h\ge 1$, a $B_h$-set is a set of integers such that every integer $n$ has at most one representation in the form $n=a_{i_1}+\cdots +a_{i_h}$, where $a_{i_r}\in A$ for all $r=1,\dots ,h$ and $a_{i_1}\le \dots \le a_{i_h}$. The greedy $B_h$-set is the infinite set of nonnegative integers $\{a_0(h),a_1(h),a_2(h),\dots \}$ constructed as follows: If $a_0(h)=0$ and $\{a_0(h),a_1(h),a_2(h),\dots ,a_k(h)\}$ is a $B_h$-set, then $a_{k+1}(h)$ is the least positive integer such that $\{a_0(h),a_1(h),a_2(h),\dots ,a_k(h),a_{k+1}(h)\}$ is a $B_h$-set. Then $a_1(h)=1$, $a_2(h)=h+1$, and $a_3(h)=h^2+h+1$ for all $h$. This paper proves that $a_4(h)$, the fourth term of the greedy $B_h$-set is $\left(h^3+3h^2+3h+1\right)/2$ if $h$ is odd and $\left(h^3+2h^2+3h+2\right)/2$ if $h$ is even.

7 pages

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