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A230100 Numbers that can be expressed as (m + sum of digits of m) in exactly three ways. 2
10000000000001, 10000000000003, 10000000000005, 10000000000007, 10000000000009, 10000000000011, 10000000000013, 10000000000015, 10000000000102, 10000000000104, 10000000000106, 10000000000108, 10000000000110, 10000000000112, 10000000000114, 10000000000116 (list; graph; refs; listen; history; text; internal format)



Let f(n) = n + (sum of digits of n) = A062028(n).

Let g(m) = number of n such that f(n) = m (i.e. the number of inverses of m), A230093(m).

Numbers m with g(m) = 0 are called the Self or Colombian numbers, A003052.

Numbers m with g(m) = 1 give A225793.

Numbers m with g(m) = 2 give A230094.

The present sequence gives numbers m such that A230093(m) = 3.

The smallest term, a(1) = 10^13 + 1, was found by Narasinga Rao, who reports that Kaprekar verified that it is the smallest term. No details of Kaprekar's proof were given.

a(2) onwards were computed by Donovan Johnson, Oct 12 2013, and on Oct 20 2013 he completed a search of all numbers below 10^13 and verified that 10^13 + 1 is indeed the smallest term.

See A006064 for much more about this question.


Joshi, V. S. A note on self-numbers. Volume dedicated to the memory of V. Ramaswami Aiyar. Math. Student 39 (1971), 327--328 (1972). MR0330032 (48 #8371)

D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.

Andrzej Makowski, On Kaprekar's "junction numbers". Math. Student 34 1966 77 (1967). MR0223292 (36 #6340)

Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)


Table of n, a(n) for n=1..16.

D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]

Index entries for Colombian or self numbers and related sequences


There are exactly three numbers, 9999999999892, 9999999999901 and 10000000000000, whose image under n->f(n) is 10000000000001, so 10^13+1 is a member of the sequence.


Cf. A006064, A062028, A230093.

Sequence in context: A175272 A104833 A112432 * A122966 A288264 A127225

Adjacent sequences: A230097 A230098 A230099 * A230101 A230102 A230103




N. J. A. Sloane, Oct 12 2013 - Oct 25 2013



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Last modified November 29 18:13 EST 2022. Contains 358431 sequences. (Running on oeis4.)