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selfness

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

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:

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 C_1 = b-1 for even bases and 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.

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

In October 2006 Luke Pebody demonstrated that the largest known Mersenne prime that is at the same time a self number is 224036583-1. 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:

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 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

Effective test

Kaprekar demonstrated that:

n mbox{ is self if } [n - DR*(n) - 9 cdot i ] + SOD([n - DR*(n) - 9 cdot i ] ) neq n quad forall i in 0 ldots d(n)

Where:

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

begin{align} DR(n) &{}= begin{cases} 9, & mbox{if } SOD(n) mod 9 = 0 SOD(n) mod 9, & mbox{ otherwise} end{cases} &{}= (n - 1) mod 9 + 1 end{align}

SOD(n) mbox{ is the sum of all digits in } n

d(n) mbox{ is the number of digits in } n

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

The following table was calculated in 2007.

BaseCertificateSum of digits
401959 = [1, 8, 39]_{40}48
41--
421967 = [1, 4, 35]_{42}40
43--
441971 = [1, 0, 35]_{44}36
441928 = [43, 36]_{44}79
45--
461926 = [41, 40]_{46}81
47--
48--
49--
501959 = [39, 9]_{50}48
51--
521947 = [37, 23]_{52}60
53--
541931 = [35, 41]_{54}76
55--
561966 = [35, 6]_{56}41
57--
581944 = [33, 30]_{58}63
59--
601918 = [31, 58]_{60}89

References

  • 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

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