Innumber theory, aself number orDevlali number in a givennumber base${\displaystyle b}$ is anatural number that cannot be written as the sum of any other natural number${\displaystyle n}$ and the individual digits of${\displaystyle n}$. 20 is a self number (in base 10), because no such combination can be found (all give a result less than 20; all other${\displaystyle n}$ give a result greater than 20). 21 is not, because it can be written as 15 + 1 + 5 usingn = 15. These numbers were first described in 1949 by theIndianmathematicianD. R. Kaprekar.^[1]

Let${\displaystyle n}$ be a natural number. We define the${\displaystyle b}$-self function for base${\displaystyle b>1}$${\displaystyle F_b:\mathbb{N}\to \mathbb{N}}$ to be the following:

${\displaystyle F_b(n)=n+\sum _{i=0}^{k-1}{d}_i.}$

where${\displaystyle k=\lfloor {\log }_{b}n\rfloor +1}$ is the number of digits in the number in base${\displaystyle b}$, and

${\displaystyle d_i=\frac{nmod{b}^{i+1}-n{modb}^i}{b^i}}$

is the value of each digit of the number. A natural number${\displaystyle n}$ is a${\displaystyle b}$-self number if thepreimage of${\displaystyle n}$ for${\displaystyle F_b}$ is theempty set.

In general, for even bases, allodd 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${\displaystyle b}$ is infinite and has a positiveasymptotic density: when${\displaystyle b}$ is odd, this density is 1/2.^[3]

Forbase 2 self numbers, seeOEIS: A010061. (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, ... (sequenceA003052 in theOEIS)

Aself prime is a self number that isprime.

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, ... (sequenceA006378 in theOEIS)

• Kaprekar, D. R.The Mathematics of New Self-Numbers Devaiali (1963): 19 - 20.
• R. B. Patel (1991). "Some Tests fork-Self Numbers".Math. Student.56:206–210.
• B. Recaman (1974). "Problem E2408".Amer. Math. Monthly.81 (4): 407.doi:10.2307/2319017.JSTOR 2319017.
• Sándor, Jozsef; Crstici, Borislav (2004).Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36.ISBN 1-4020-2546-7.Zbl 1079.11001.
• Weisstein, Eric W."Self Number".MathWorld.

• Arithmetic dynamics
• Base-dependent integer sequences
• Inverse functions

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