cp's OEIS Frontend

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

%I A230100 #76 Apr 24 2025 13:18:05
%S A230100 10000000000001,10000000000003,10000000000005,10000000000007,
%T A230100 10000000000009,10000000000011,10000000000013,10000000000015,
%U A230100 10000000000102,10000000000104,10000000000106,10000000000108,10000000000110,10000000000112,10000000000114,10000000000116
%N A230100 Numbers that can be expressed as (m + sum of digits of m) in exactly three ways.
%C A230100 Let f(n) = n + (sum of digits of n) = A062028(n).
%C A230100 Let g(m) = number of n such that f(n) = m (i.e. the number of inverses of m), A230093(m).
%C A230100 Numbers m with g(m) = 0 are called the Self or Colombian numbers, A003052.
%C A230100 Numbers m with g(m) = 1 give A225793.
%C A230100 Numbers m with g(m) = 2 give A230094.
%C A230100 The present sequence gives numbers m such that A230093(m) = 3.
%C A230100 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.
%C A230100 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.
%C A230100 See A006064 for much more about this question.
%C A230100 Numbers m with g(m) = 4 give A377422. - _Daniel Mondot_, Oct 29 2024
%D A230100 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 A230100 D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
%D A230100 Andrzej Makowski, On Kaprekar's "junction numbers", Math. Student 34 1966 77 (1967). MR0223292 (36 #6340)
%D A230100 A. Narasinga Rao, On a technique for obtaining numbers with a multiplicity of generators, Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)
%H A230100 Daniel Mondot, <a href="/A230100/b230100.txt">Table of n, a(n) for n = 1..10000</a>
%H A230100 Max A. Alekseyev and N. J. A. Sloane, <a href="https://arxiv.org/abs/2112.14365">On Kaprekar's Junction Numbers</a>, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
%H A230100 D. R. Kaprekar, <a href="/A003052/a003052_2.pdf">The Mathematics of the New Self Numbers</a> [annotated and scanned]
%H A230100 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%e A230100 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.
%Y A230100 Cf. A006064, A062028, A230093.
%K A230100 nonn,base
%O A230100 1,1
%A A230100 _N. J. A. Sloane_, Oct 12 2013 - Oct 25 2013