Function Repository Resource:

# SmallestPartsFunction

Total number of smallest parts in the partitions of a positive integer

Contributed by: George Beck
 ResourceFunction["SmallestPartsFunction"][n] gives the total number of smallest parts in the integer partitions of the positive integer n.

## Details and Options

A partition of n is a list of weakly decreasing positive integers that add up to n. For instance, written compactly, these are the five partitions of 4: 4, 31, 22, 211, 1111.

## Examples

### Basic Examples

Here is an illustrative example:

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The integer partitions of 3:

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The smallest parts:

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The number of smallest parts per partition:

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Their total:

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Here are the first 10 numbers in the smallest part function (spt) sequence:

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### Neat Examples

The function spt(n) satisfies some congruences similar to Ramanujan’s congruences for the partition function p(n).

Numbers of the form 5n+4 have spt divisible by 5:

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Numbers of the form 7n+5 have spt divisible by 7:

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Numbers of the form 13n+6 have spt divisible by 13:

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