Self number

A self number, Colombian number or Devlali number is an integer which, in a given base, cannot be generated by any other integer added to the sum of its digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is, 21 = 15 + 1 + 5. No such sum will generate the integer 20, hence it is a self number. These numbers were first described in 1949 by the Indian mathematician D. R. Kaprekar.

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, 501, 512, 514, 525 (sequence A003052 in the OEIS)

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 following recurrence relation generates base 10 self numbers:

${\displaystyle C_{k}=8\cdot 10^{k-1}+C_{k-1}+8}$

(with C₁ = 9)

And for binary numbers:

${\displaystyle C_{k}=2^{j}+C_{k-1}+1\,}$

(where j stands for the number of digits) we can generalize a recurrence relation to generate self numbers in any base b:

${\displaystyle C_{k}=(b-2)b^{k-1}+C_{k-1}+(b-2)\,}$

in which ${\displaystyle C_{1}=b-1}$ for even bases and ${\displaystyle C_{1}=b-2}$ for odd bases.

The existence of these recurrence relations shows that for any base there are infinitely many self numbers.

A search for self numbers can turn up self-descriptive numbers, which are similar to self numbers in being base-dependent, but quite different in definition and much fewer in frequency.

A self prime is a self number that is prime. The first few self primes (sequence A006378 in the OEIS) are

3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389

In October 2006 Luke Pebody demonstrated that the largest known Mersenne prime that is at the same time a self number is 2^24036583-1. This is then the largest known self prime as of 2006.

Luke Pebody showed (Oct 2006) that a link can be made between the self property of a large number n and a low-order portion of that number, adjusted for digit sums:

a) In general, n is self if and only if m = R(n)+SOD(R(n))-SOD(n) is self

Where:

R(n) is the smallest rightmost digits of n, greater than 9.d(n)

d(n) is the number of digits in n

SOD(x) is the sum of digits of x, the function ${\displaystyle S_{10}(x)}$ from above.

b) If n = a.10^b+c, c<10^b, then n is self if and only if both {m1 & m2} are negative or self

Where:

m1 = c - SOD(a)

m2 = SOD(a-1)+9.b-(c+1)

c) For the simple case of a=1 & c=0 in the previous model (i.e. n=10^b), then n is self if and only if (9.b-1) is self

Kaprekar demonstrated that:

n is self if [n-DR*(n)-9.i] + SOD([n-DR*(n)-9.i]) ≠ n for any 0 <= i <= d(n)

Where:

DR*(n) is equal to DR(n)/2 if DR(n) is even, otherwise is equal to (DR(n)+9)/2

DR(n) is equal to SOD(n) mod 9, or equal to 9 if SOD(n) mod 9 = 0

The following table was calculated in 2007.

Base	Certificate	Sum of digits
40	${\displaystyle 1959=[1,8,39]_{40}}$	48
41	-	-
42	${\displaystyle 1967=[1,4,35]_{42}}$	40
43	-	-
44	${\displaystyle 1971=[1,0,35]_{44}}$	36
44	${\displaystyle 1928=[43,36]_{44}}$	79
45	-	-
46	${\displaystyle 1926=[41,40]_{46}}$	81
47	-	-
48	-	-
49	-	-
50	${\displaystyle 1959=[39,9]_{50}}$	48
51	-	-
52	${\displaystyle 1947=[37,23]_{52}}$	60
53	-	-
54	${\displaystyle 1931=[35,41]_{54}}$	76
55	-	-
56	${\displaystyle 1966=[35,6]_{56}}$	41
57	-	-
58	${\displaystyle 1944=[33,30]_{58}}$	63
59	-	-
60	${\displaystyle 1918=[31,58]_{60}}$	89

• Kaprekar, D. R. The Mathematics of New Self-Numbers Devaiali (1963): 19 - 20.
• Patel, R. B. "Some Tests for -Self Numbers" Math. Student 56 (1991): 206 - 210.
• B. Recaman, "Problem E2408" Amer. Math. Monthly 81 (1974): 407
• Weisstein, Eric W. "Self Number". MathWorld.