Consecutive numbers (or more properly, consecutive integers) are integers 
and 
such that 
, i.e., 
follows immediately after 
.
Given two consecutive numbers, one must be even and one must be odd. Since the product of an even number and an odd number is always even, the product of two consecutive numbers (and, in fact, of any two or more consecutive numbers) is always even.
Marotti and Needleman (2026) considered sums of one or more consecutive numbers in the cyclic list 1, 2, ..., 
, allowing the sum to wrap from 
to 1 and using each entry at most once.
They proved that every integer from 1 through the triangular number 
is representable iff 
, 2, 3, 4, 5, or 7. For 
, the values 8 and 13 are missing, and every 
has at least one missing value. For example, when 
, the sum 
is permitted.
