Self number

Definition and properties

Let n be a natural number. We define the b-self function F_b : \mathbb{N} \rightarrow \mathbb{N} for base b 1 to be the following: :F_{b}(n) = n + \sum_{i=0}^{k - 1} d_i. where k = \lfloor \log_{b}{n} \rfloor + 1 is the number of digits in the number in base b, and :d_i = \frac{n \bmod{b^{i+1}} - n \bmod b^i}{b^i} is the value of each digit of the number. A natural number n is a b-self number if the preimage of n for F_b is the empty set.

In general, for even bases, all odd numbers below the base number are self numbers, since any number below such an odd number would have to also be a 1-digit number which when added to its digit would result in an even number. For odd bases, all odd numbers are self numbers.

The set of self numbers in a given base b is infinite and has a positive asymptotic density: when b is odd, this density is 1/2.

Self numbers in specific bases

For base 2 self numbers, see . (written in base 10)

The first few base 10 self numbers are: : 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, ...

Self primes

A self prime is a self number that is prime.

The first few self primes in base 10 are :3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873, ...

References

• Kaprekar, D. R. The Mathematics of New Self-Numbers Devaiali (1963): 19 - 20.

References

1. {{OEIS
2. Sándor & Crstici (2004) p.384
3. Sándor & Crstici (2004) p.385