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:

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

This sequence is A003635 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Historical Note

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