On the Sum of Consecutive Integers in Sequences Ii

Abstract

Let A be a sequence of natural numbers, r A (n) be the number of ways to represent n as a sum of consecutive elements in A, and M A (x) := P n≤x r A (n). We give a new short proof of LeVeque's formula regarding M A (x) when A is an arithmetic progression, and then extend the proof to give asymptotic formulas for the case when A behaves almost like an arithmetic progression, and also when A is the set of primes in an arithmetic progression.

Trapezoidal numbers, divisor functions, and a partition theorem of Sylvester

“…. , s ′ we have 28) is a maximal sequence of consecutive integers in the major-minor hook partition of n. It follows that the number of pairwise disjoint maximal sequences of consecutive integers in the major-minor hook partition of n is exactly one more than the number of pairwise disjoint maximal sequences of consecutive integers in the major-minor hook partition of m. By the induction hypothesis, the latter partition consists of ℓ − 1 maximal disjoint sequences, and so the partition ( 25 Because j < r j , the major-minor hook partition of m satisfies the relations ( 21), ( 22), (23), and (24). The major and minor halves of the center-justified Ferrers diagram for the partition (19) of n also have Durfee squares with side j.…”

“…For other recent work on trapezoidal numbers, see Apostol [8], Guy [12], Leveque [14], Moser [17], Pong [18,19], and Tsai and Zaharescu [23,24].…”