A006064 Smallest junction number with n generators. (Formerly M5367)

0, 101, 10000000000001, 1000000000000000000000102

Offset

1,2

Comments

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

References

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).

Links

Table of n, a(n) for n=1..4.

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, 2022 (to appear).

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 [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]

Index entries for Colombian or self numbers and related sequences

Formula

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

Example

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.

Crossrefs

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.

Sequence in context: A138720 A262627 A259199 * A230304 A015164 A325776

Adjacent sequences: A006061 A006062 A006063 * A006065 A006066 A006067

Keyword

nonn,base

Author

N. J. A. Sloane.

Extensions

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

Status

approved