|
%I #68 Sep 04 2022 14:43:17
%S 101,103,105,107,109,111,113,115,117,202,204,206,208,210,212,214,216,
%T 218,303,305,307,309,311,313,315,317,319,404,406,408,410,412,414,416,
%U 418,420,505,507,509,511,513,515,517,519,521,606,608,610,612,614,616,618,620,622,707,709,711,713,715,717,719,721,723,808
%N Numbers that can be expressed as (m + sum of digits of m) in exactly two ways.
%C Numbers n such that A230093(n) = 2.
%C The sequence "Numbers n such that A230093(n) = 3" starts at 10^13+1 (see A230092). This implies that changing the definition of A230094 to "Numbers n such that A230093(n) >= 2" (the so-called "junction numbers") would produce a sequence which agrees with A230094 up to 10^13.
%C Makowski shows that the sequence of junction numbers is infinite.
%D Joshi, V. S. 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 D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
%D D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.
%D Makowski, Andrzej. On Kaprekar's "junction numbers''. Math. Student 34 1966 77 (1967). MR0223292 (36 #6340)
%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)
%H Reinhard Zumkeller, <a href="/A230094/b230094.txt">Table of n, a(n) for n = 1..10000</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 Santanu Bandyopadhyay, <a href="https://www.ese.iitb.ac.in/~santanu/RM8.pdf">Self-Number</a>, Indian Institute of Technology Bombay (Mumbai, India, 2020).
%H Santanu Bandyopadhyay, <a href="/A003052/a003052_3.pdf">Self-Number</a>, Indian Institute of Technology Bombay (Mumbai, India, 2020). [Local copy]
%H David A. Corneth, <a href="/A230094/a230094.gp.txt">Examples</a>
%H D. R. Kaprekar, <a href="/A003052/a003052_2.pdf">The Mathematics of the New Self Numbers</a> [annotated and scanned]
%H <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%e a(1) = 101 = 91 + (9+1) = 100 + (1+0+0);
%e a(10) = 202 = 191 + (1+9+1) = 200 + (2+0+0);
%e a(100) = 1106 = 1093 + (1+0+9+3) = 1102 + (1+1+0+2);
%e a(1000) = 10312 = 10295 + (1+0+2+9+5) = 10304 + (1+0+3+0+4).
%p For Maple code see A230093.
%t Position[#, 2][[All, 1]] - 1 &@ Sort[Join[#2, Map[{#, 0} &, Complement[Range[#1], #2[[All, 1]]]] ] ][[All, -1]] & @@ {#, Tally@ Array[# + Total@ IntegerDigits@ # &, # + 1, 0]} &[10^3] (* _Michael De Vlieger_, Oct 28 2020, after _Harvey P. Dale_ at A230093 *)
%o (Haskell)
%o a230094 n = a230094_list !! (n-1)
%o a230094_list = filter ((== 2) . a230093) [0..]
%o -- _Reinhard Zumkeller_, Oct 11 2013
%Y Cf. A003052, A007953, A004207, A048528, A062028, A176995, A225793, A227915, A230092, A230093.
%K nonn,base
%O 1,1
%A _N. J. A. Sloane_, Oct 10 2013, Oct 24 2013
|
