Abstract
It is known, though perhaps not as well as it should be, that the number of partitions of n into (one or more) consecutive parts is equal to the number of odd divisors of n. (This is the special case k = 1 of a theorem of J. J. Sylvester [1, §46], to the effect that the number of partitions of n into distinct parts with k sequences of consecutive parts is equal to the number of partitions of n into odd parts (repetitions allowed) precisely k of which are distinct.) For instance,
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