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%I M5367 #79 Jan 13 2023 17:59:29
%S 0,101,10000000000001,1000000000000000000000102
%N Smallest junction number with n generators.
%C Strictly speaking, a junction number is a number n with more than one solution to x+digitsum(x) = n. However, it seems best to start this sequence with n=0, for which there is just one solution, x=0. - _N. J. A. Sloane_, Oct 31 2013.
%C a(3) = 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 a(4) = 10^24 + 102 was conjectured by Narasinga Rao.
%C a(5) = 10^1111111111124 + 102. - Conjectured by Narasinga Rao, confirmed by _Max Alekseyev_ and _N. J. A. Sloane_.
%C a(6) = 10^2222222222224 + 10000000000002. - _Max Alekseyev_
%C a(7) = 10^( (10^24 + 10^13 + 115) / 9 ) + 10^13 + 2. - _Max Alekseyev_
%C a(8) = 10^( (2*10^24 + 214)/9 ) + 10^24 + 103. - _Max Alekseyev_
%D M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
%D D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
%D Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Max Alekseyev, <a href="/A006064/a006064.txt">Table of expressions for a(n), for n=1..100</a>
%H 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, 2022 (to appear).
%H D. R. Kaprekar, <a href="/A003052/a003052_2.pdf">The Mathematics of the New Self Numbers</a> [annotated and scanned]
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 21.
%H Terry Trotter, <a href="http://trottermath.net/charlene-numbers/">Charlene numbers</a> [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here. - _N. J. A. Sloane_, Mar 29 2018]
%H <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%F a(n) = the smallest m such that there are exactly n solutions to A062028(x)=m.
%e a(2) = 101 since 101 is the smallest number with two generators: 101 = A062028(91) = A062028(100).
%e a(4) = 10^24 + 102 = 1000000000000000000000102 has exactly four inverses w.r.t. A062028, namely 999999999999999999999893, 999999999999999999999902, 1000000000000000000000091 and 1000000000000000000000100.
%Y Cf. A003052, A230093, A230100, A230303, A230857 (highest power of 10).
%Y Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.
%K nonn,base
%O 1,2
%A _N. J. A. Sloane_.
%E Edited, a(5)-a(6) added by _Max Alekseyev_, Jun 01 2011
%E a(1) added, a(5) corrected, a(7)-a(8) added by _Max Alekseyev_, Oct 26 2013
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