A230100 Numbers that can be expressed as (m + sum of digits of m) in exactly three ways.

10000000000001, 10000000000003, 10000000000005, 10000000000007, 10000000000009, 10000000000011, 10000000000013, 10000000000015, 10000000000102, 10000000000104, 10000000000106, 10000000000108, 10000000000110, 10000000000112, 10000000000114, 10000000000116

Offset: 1

Author

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

Keywords

• nonn
• base

Comments

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.

Numbers m with g(m) = 4 give A377422. - Daniel Mondot, Oct 29 2024

Examples

			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.
		

References

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

Links

• Daniel Mondot, Table of n, a(n) for n = 1..10000
• Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
• D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A006064, A062028, A230093.