Self number explained

is a natural number that cannot be written as the sum of any other natural number

and the individual digits of

. 20 is a self number (in base 10), because no such combination can be found (all

n<15

give a result less than 20; all other

give a result greater than 20). 21 is not, because it can be written as 15 + 1 + 5 using n = 15. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar.^[1]

Definition and properties

Let

be a natural number. We define the

-self function for base

b>1

F_b:N → N

to be the following:

F_b(n)=n+

	k-1
\sum
	i=0

d_i.

where

k=\lfloorlog_b{n}\rfloor+1

is the number of digits in the number in base

, and

d_i=

	n\bmod{bⁱ⁺¹
	-

n\bmodb^i}{b^i}

is the value of each digit of the number. A natural number

is a

-self number if the preimage of

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.^[2]

The set of self numbers in a given base

is infinite and has a positive asymptotic density: when

is odd, this density is 1/2.^[3]

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.
R. B. Patel . Some Tests for k-Self Numbers . Math. Student . 56 . 1991 . 206–210.
B. Recaman . Problem E2408 . Amer. Math. Monthly . 81 . 4 . 1974 . 407 . 10.2307/2319017. 2319017 .
Book: Sándor . Jozsef . Crstici . Borislav . Handbook of number theory II . Dordrecht . Kluwer Academic . 2004 . 1-4020-2546-7 . 32–36 . 1079.11001 .

Notes and References

Web site: Self Numbers . James P.. Curley. April 30, 2015. 2024-02-29.
Sándor & Crstici (2004) p.384
Sándor & Crstici (2004) p.385