Numeroperustan

Numeroperustan is a term used in number theory to denote a positive integer whose decimal digits satisfy a simple invariant: the sum of its digits equals the product of its digits.

Properties: Since a zero digit would force the product to be zero while the sum is positive, numeroperustans necessarily have no zero digits. Digits are from 1 to 9. The simplest numeroperustan are the single-digit numbers 1, 2, and 3, which all satisfy the invariant. Beyond that, there are many numbers; for example 22 (sum 4, product 4) and 123 (sum 6, product 6). In general, consider numbers formed with only 1s and 2s. If the number has q ones and p twos, the condition becomes q + 2p = 2^p, or q = 2^p - 2p. This yields infinitely many solutions with p ≥ 1 (for instance p=3 gives q=2, so any arrangement of two 1s and three 2s, such as 11222, is a numeroperustan).

Examples: 1, 2, 3, 22, 123, 132, 213, 1124, 11222, and longer strings constructed from the 1s-and-2s pattern.

Relation to other concepts: Numeroperustan is discussed in recreational mathematics contexts as a digit-based invariant. It differs from Harshad (Niven) numbers, which require divisibility by the sum of digits, rather than equality with the product.

See also: Harshad numbers, digit sum, multiplicative digital invariant.