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# Arithmetic progressions

-terms **arithmetic progressions** are sequences of the form (with for an infinity of terms)

where and are constants; therefore and . For example, {4, 16, 28, 40, 52, 64, 76, 88, 100, 112, ...} (A017569) is an arithmetic progression with and . In terms of growth of sequences, nonconstant arithmetic progressions have linear growth.

Equivalently, a sequence is an arithmetic progression when each term is the arithmetic mean of the neighboring terms, i.e.

## Contents

## "Primitive" versus "nonprimitive" arithmetic progressions

An arithmetic progression might be said to be "primitive" if and are coprime. An arithmetic progression with (cf. gcd), which might thus be said to be "nonprimitive", is times the corresponding "primitive" arithmetic progression. For example {4, 16, 28, 40, 52, 64, 76, 88, 100, 112, ...} = 4 × {1, 4, 7, 10, 13, 16, 19, 22, 25, 28, ...}.

## Recurrence

## Generating functions

Arithmetic progressions have rational [ordinary] generating functions of the form

## See also

- Dirichlet's theorem on arithmetic progressions
- Primes in arithmetic progression
- Consecutive primes in arithmetic progression