Self number

A self number, Colombian number or Devlali number is an integer that cannot be written as the sum of any other integer n and the individual digits of n. This property is specific to the base used to represent the integers. 20 is a self number (in base 10), because no such combination can be found (all n < 15 give a result < 20; all other n give a result > 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.

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, ... (sequence A003052 in the OEIS)

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.

Properties

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.^[1]

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

Recurrent formula

The following recurrence relation generates some base 10 self numbers:

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

(with C₁ = 9)

And for binary numbers:

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:

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

in which C₁ = b − 1 for even bases and C₁ = b − 2 for odd bases.

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

Self primes

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

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.

Selfness tests

Reduction tests

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:

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 S₁₀(x) from above.
If $n=a\cdot 10^{b}+c,\ c<10^{b}$ , 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)
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

Effective test

Kaprekar demonstrated that:

n

is self if

\mathrm {SOD} (|n-\mathrm {DR} ^{*}(n)-9\cdot i|)\neq [\mathrm {DR} ^{*}(n)+9\cdot i]\quad \forall i\in 0\ldots d(n)

Where:

\mathrm {DR} ^{*}(n)={\begin{cases}{\frac {\mathrm {DR} (n)}{2}},&{\text{if }}\mathrm {DR} (n){\text{ is even}}\\{\frac {\mathrm {DR} (n)+9}{2}},&{\text{if }}\mathrm {DR} (n){\text{ is odd}}\end{cases}}

{\begin{aligned}\mathrm {DR} (n)&{}={\begin{cases}9,&{\text{if }}\mathrm {SOD} (n)\mod 9=0\\\mathrm {SOD} (n)\mod 9,&{\text{otherwise}}\end{cases}}\\&{}=1+[(n-1)\mod 9]\end{aligned}}

\mathrm {SOD} (n)

is the sum of all digits in

n

.

d(n)

is the number of digits in

n

.

Other bases

In base 12, the self numbers are: (using inverted two and three for ten and eleven, respectively)

1, 3, 5, 7, 9, Ɛ, 20, 31, 42, 53, 64, 75, 86, 97, ᘔ8, Ɛ9, 10ᘔ, 110, 121, 132, 143, 154, 165, 176, 187, 198, 1ᘔ9, 1Ɛᘔ, 20Ɛ, 211, 222, 233, 244, 255, 266, 277, 288, 299, 2ᘔᘔ, 2ƐƐ, 310, 312, 323, 334, 345, 356, 367, 378, 389, 39ᘔ, 3ᘔƐ, 400, 411, 413, 424, 435, 446, 457, 468, 479, 48ᘔ, 49Ɛ, 4Ɛ0, 501, 512, 514, 525, 536, 547, 558, 569, 57ᘔ, 58Ɛ, 5ᘔ0, 5Ɛ1, ...

The self primes are: (using inverted two and three for ten and eleven, respectively)

3, 5, 7, Ɛ, 31, 75, 255, 277, 2ƐƐ, 3ᘔƐ, 435, 457, 58Ɛ, 5Ɛ1, ...

For base 2 self numbers, see A010061. (written in base 10)

Excerpt from the table of bases where 2007 is self or Colombian

The following table was calculated in 2007.

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

References

↑ Sándor & Crstici (2004) p.384
↑ Sándor & Crstici (2004) p.385

Kaprekar, D. R. The Mathematics of New Self-Numbers Devaiali (1963): 19 - 20.
R. B. Patel (1991). "Some Tests for k-Self Numbers". Math. Student. 56: 206–210.
B. Recaman (1974). "Problem E2408". Amer. Math. Monthly. 81 (4): 407. doi:10.2307/2319017.
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.

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[CS384-1] Sándor & Crstici (2004) p.384

[CS385-2] Sándor & Crstici (2004) p.385

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