Self Number

A number (usually base 10 unless specified otherwise) which has no digitaddition generator. Such numbers were originally called Colombian numbers (S. 1974). There are infinitely many such numbers, since an infinite sequence of self numbers can be generated from the recurrence relation


for k=2, 3, ..., where C_1=9. The first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, ... (OEIS A003052).

An infinite number of 2-self numbers (i.e., base-2 self numbers) can be generated by the sequence


for k=1, 2, ..., where C_1=1 and j is the number of digits in C_(k-1). An infinite number of n-self numbers can be generated from the sequence


for k=2, 3, ..., and

 C_1={n-1   for n even; n-2   for n odd.

Joshi (1973) proved that if k is odd, then m is a k-self number iff m is odd. Patel (1991) proved that 2k, 4k+2, and k^2+2k+1 are k-self numbers in every even base k>=4.

See also


Explore with Wolfram|Alpha


Cai, T. "On k-Self Numbers and Universal Generated Numbers." Fib. Quart. 34, 144-146, 1996.Gardner, M. Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 115-117, 122, 1988.Joshi, V. S. Ph.D. dissertation. Gujarat University, Ahmadabad, 1973.Kaprekar, D. R. The Mathematics of New Self-Numbers. Devaiali, pp. 19-20, 1963.Patel, R. B. "Some Tests for k-Self Numbers." Math. Student 56, 206-210, 1991.S., B. R. "Solution to Problem E 2048." Amer. Math. Monthly 81, 407, 1974.Sloane, N. J. A. Sequence A003052/M2404 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Self Number

Cite this as:

Weisstein, Eric W. "Self Number." From MathWorld--A Wolfram Web Resource.

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