SUPER SMITH–NIVEN NUMBERS



Definition & Examples
We observe that the number 666—also known as the beast number [1]—obeys three peculiar digital properties:
  1. The digital sum 6 + 6 + 6 is a factor of 666, as we see from the factorization 666 = 18 × 37.
  2. The quotient 37, from the division 666 ÷ (6 + 6 + 6), is a prime number, i.e., a number which cannot be factored any further.
  3. The complete factorization 666 = 2 × 3 × 3 × 37 yields the same digital sum, i.e., 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7.
A number with Property (1) is known as a Niven number, and a number satisfying Property (3) is called a Smith number. Accordingly, we say that a number is super Smith–Niven [2], abbreviated SSN, if it enjoys all the three features shared by the number 666. With this definition, we find that the smallest SSN number is 27, for which Properties (1,2,3) hold as 27 = (2 + 7) × 3 = 3 × 3 × 3 and 2 + 7 = 3 + 3 + 3.
[1] The Holy Bible, Revelation XIII v. 18 [KingJames]
[2] Journal of Combinatorics and Number Theory 5 (2013) pp. 215–225 [MathSciNet]
Super Digital Sums
In particular, the beast number itself places third in the sequence of SSN numbers: 27, 645, 666, which is then succeeded by twelve more numbers before the terms exceed the ten-thousand range:

915, 1962, 2265, 2286, 2934, 3258, 3345, 3615, 5526, 6315, 9015, 9414, …

Notice that all these numbers have digital sums either 15 or 18. This is a fact, that SSN numbers cannot have an arbitrary digital sum. Except for 27, which has digital sum 9, the digital sums belonging to SSN numbers are limited to a certain list [2: Table 2]—albeit neverending—which begins as follows.

15, 18, 33, 36, 45, 48, 51, 54, 63, 69, 84, 87, 90, 98, 99, …


Fig. 1: Three successive SSN numbers showing on the odometer with their respective digital sum.

Observation Tables
We conjecture that SSN numbers are infinitely many, although they are relatively rare. For instance, in the interval up to a billion, we identify only about 150 thousand SSN numbers—that is an average of one SSN in every 6666 natural numbers!

Now one can ask many stimulating questions that call for further investigation on SSN numbers, e.g., where their distribution is concerned. For research purposes, we have made available a collection of computational output in the following files.
  1. The preceding list of super digital sums, extended up to 104 (txt file).
  2. The list of all SSN numbers up to 107 (txt file) and up to 109 (zip file 0.5 Mb).
  3. The list of SSN primes, i.e., the prime quotients from Property (2), compiled up to 106 (txt file) and up to 109 (zip file 10 Mb).
This page was created on 20.01.2014 and last updated on 30.01.2017.
Copyright © 2014–2017 Amin Witno