A PARTITION PROBLEM

[摘要] For positive integers l, n, k we say that M = M(n, k) = {n, n + 1, ..., n + k} has an l-partition, if there is A subset-of M(n, k) with lSIGMA(a is-an-element-of A) a = SIGMA(m is-an-element-of M)m. Moreover, define B(l) = {M(n, k):M has an l-partition}, and K(l) = min(M is-an-element-of B(l)) (\M| - 1). We show that M(n, k) is-an-element-of B(2) iff k = 3 mod 4 or 2\k,n = k/2 mod 2, 4n less-than-or-equal-to k2. Then we prove an explicit formula for K(p(d)), where p is prime; finally, we introduce a method of determining K(r) for arbitrary r is-an-element-of N, particularly for r = pq with primes p and q. (C) 1994 Academic Press, Inc.

[发布日期] 1994-08-01