Glaisher's theorem

In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. It is named for James Whitbread Lee Glaisher.

It states that the number of partitions of an integer $N$ into parts not divisible by $d$ is equal to the number of partitions of the form

$N=N_1+\cdots+N_k$

where $N_i\geq N_{i+1}$ and $N_i\geq N_{i+d-1}+1$ .

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