A230100 -id:A230100 - cp's OEIS frontend

A006064 Smallest junction number with n generators.

0, 101, 10000000000001, 1000000000000000000000102

Offset: 1

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.
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.
a(4) = 10^24 + 102 was conjectured by Narasinga Rao.
a(5) = 10^1111111111124 + 102. - Conjectured by Narasinga Rao, confirmed by Max Alekseyev and N. J. A. Sloane.
a(6) = 10^2222222222224 + 10000000000002. - Max Alekseyev
a(7) = 10^( (10^24 + 10^13 + 115) / 9 ) + 10^13 + 2. - Max Alekseyev
a(8) = 10^( (2*10^24 + 214)/9 ) + 10^24 + 103. - Max Alekseyev

			a(2) = 101 since 101 is the smallest number with two generators: 101 = A062028(91) = A062028(100).
a(4) = 10^24 + 102 = 1000000000000000000000102 has exactly four inverses w.r.t. A062028, namely 999999999999999999999893, 999999999999999999999902, 1000000000000000000000091 and 1000000000000000000000100.

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of expressions for a(n), for n=1..100
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.
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
Terry Trotter, Charlene numbers
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A230093, A230100, A230303, A230857 (highest power of 10).

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.

a(n) = the smallest m such that there are exactly n solutions to A062028(x)=m.

Edited, a(5)-a(6) added by Max Alekseyev, Jun 01 2011

a(1) added, a(5) corrected, a(7)-a(8) added by Max Alekseyev, Oct 26 2013

A377422 Numbers that can be expressed as (m + sum of digits of m) in exactly four ways.

Original entry on oeis.org

1000000000000000000000102, 1000000000000000000000104, 1000000000000000000000106, 1000000000000000000000108, 1000000000000000000000110, 1000000000000000000000112, 1000000000000000000000114, 2000000000000000000000103, 2000000000000000000000105, 2000000000000000000000107, 2000000000000000000000109

Offset: 1

Daniel Mondot, Oct 27 2024

Numbers k such that A230093(k) = 0 give A003052, the Self or Colombian numbers.
Numbers k such that A230093(k) = 1 give A225793.
Numbers k such that A230093(k) = 2 give A230094.
Numbers k such that A230093(k) = 3 give A230100.
Numbers k such that A230093(k) = 4 give this sequence.

			There are exactly four numbers, 999999999999999999999894, 999999999999999999999903, 1000000000000000000000092, and 1000000000000000000000101, whose image under n->f(n) is 1000000000000000000000104, so 1000000000000000000000104 is a member of the sequence.

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.

Cf. A230100, A006064, A062028, A230093.

Corrected by Daniel Mondot, Apr 17 2025

A230304 a(n) = 10^( (10^n-1)/9 + n) + 1.

Original entry on oeis.org

101, 10000000000001, 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

Offset: 1

N. J. A. Sloane, Oct 26 2013

Makowski observes that A230093(a(n)) >= 2 for all n >= 1.

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

Index entries for Colombian or self numbers and related sequences

Cf. A006064, A230093, A230094, A230100.

cp's OEIS Frontend

A006064 Smallest junction number with n generators.

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A377422 Numbers that can be expressed as (m + sum of digits of m) in exactly four ways.

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A230304 a(n) = 10^( (10^n-1)/9 + n) + 1.

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