# Definition:Inconsummate Number

## Definition

Let $m \in \Z_{>0}$ be a positive integer.

Let $s_{10}$ denote the digit sum base $10$ .

$m$ is an inconsummate number if and only if:

$\nexists n \in \Z_{>0}: n = m \times s_{10} \left({n}\right)$

That is, if and only if there exists no positive integer $n \in \Z_{>0}$ such that $n$ equals $m$ multiplied by the digit sum of $n$.

### Sequence of Inconsummate Numbers

The sequence of inconsummate numbers begins:

$62, 63, 65, 75, 84, 95, 161, 173, 195, 216, 261, 266, 272, 276, \ldots$

## Historical Note

The concept of an inconsummate number appears first to have been raised by John Horton Conway to Neil Sloane in personal correspondence.