A self number or Colombian 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 integers are commonly denoted by the above symbol. ... A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ... Year 1949 (MCMXLIX) was a common year starting on Saturday (link will display the full calendar) of the Gregorian calendar. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Shri Dattathreya Ramachandra Kaprekar (1905- 1986) was an Indian mathematician, whose name is associated with a number of concepts in number theory. ...

Remarks

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 OEIS) This article is about the number one. ... Look up three in Wiktionary, the free dictionary. ... Look up five in Wiktionary, the free dictionary. ... Seven Days of Creation - 1765 book, title page 7 (seven) is the natural number following 6 and preceding 8. ... This article is about the number. ... 20 (twenty) is the natural number following 19 and preceding 21. ... 31 (thirty-one) is the natural number following 30 and preceding 32. ... Look up forty-two in Wiktionary, the free dictionary. ... 53 (fifty-three) is the natural number following 52 and preceding 54. ... 64 (sixty-four) is the natural number following 63 and preceding 65. ... 75 (seventy-five) is the natural number following 74 and preceding 76. ... 86 (eighty-six) is the natural number following 85 and preceding 87. ... 97 is the natural number following 96 and preceding 98. ... 108 is the natural number following 107 and preceding 109. ... 110 (one hundred [and] ten) is the natural number following 109 and preceding 111. ... 121 is the natural number following 120 and preceding 122. ... 132 is the natural number following 131 and preceding 133. ... 143 is the natural number following 142 and preceding 144. ... 154 is the natural number following one hundred fifty-three and preceding one hundred fifty-six ... Cardinal One hundred [and] eighty-seven Ordinal 187th Factorization Roman numeral CLXXXVII Binary 10111011 Hexadecimal BB 187 is the natural number following 186 and preceding 188. ... 211 is the natural number between 210 and 212. ... 222 (two hundred [and] twenty-two) is the natural number following 221 and preceding 223. ... 233 is the natural number between 232 and 234. ... 255 (two hundred [and] fifty-five, CCLV) is the natural number following 254 and preceding 256. ... 277 (read as two hundred and seventy-seven) is the natural number following 276 and preceding 278. ... Four hundred is the natural number following three hundred ninety-nine and preceding four hundred one. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

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. In mathematics, the parity of an object refers to whether it is even or odd. ...

The following recurrence relation generates base 10 self numbers: In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... For other uses, see Decimal (disambiguation). ...

(with C₁ = 9)

And for binary numbers: The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ...

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

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.

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-descriptive number is an integer m that in a given base b is b-digits long in which each digit d at position n (the most significant digit being at position 0 and the least significant at position b - 1) counts how many instances of digit n are...

Self primes

A self prime is a self number that is prime. The first few self primes (sequence A006378 in OEIS) are In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

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. In mathematics, a Mersenne number is a number that is one less than a power of two. ... 2006 is a common year starting on Sunday of the Gregorian calendar. ...

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:

a) In general, n is self if and only if m = R(n)+SOD(R(n))-SOD(n) is self ↔ ⇔ ≡ logical symbols representing iff. ...

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.

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

Effective Test

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

Excerpt from the table of bases where 2007 is Colombian

The following table was calculated in 2007.