A230094 Numbers that can be expressed as (m + sum of digits of m) in exactly two ways.

101, 103, 105, 107, 109, 111, 113, 115, 117, 202, 204, 206, 208, 210, 212, 214, 216, 218, 303, 305, 307, 309, 311, 313, 315, 317, 319, 404, 406, 408, 410, 412, 414, 416, 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

Offset: 1

Author

N. J. A. Sloane, Oct 10 2013, Oct 24 2013

Keywords

• nonn
• base

Comments

Numbers n such that A230093(n) = 2.

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.

Makowski shows that the sequence of junction numbers is infinite.

Examples

			a(1) = 101 = 91 + (9+1) = 100 + (1+0+0);
a(10) = 202 = 191 + (1+9+1) = 200 + (2+0+0);
a(100) = 1106 = 1093 + (1+0+9+3) = 1102 + (1+1+0+2);
a(1000) = 10312 = 10295 + (1+0+2+9+5) = 10304 + (1+0+3+0+4).
		

References

• 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. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
• D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.
• Makowski, Andrzej. On Kaprekar's "junction numbers''. Math. Student 34 1966 77 (1967). MR0223292 (36 #6340)
• Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)

Links

• Reinhard Zumkeller, 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.
• Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020).
• Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020). [Local copy]
• David A. Corneth, Examples
• 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]
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A003052, A007953, A004207, A048528, A062028, A176995, A225793, A227915, A230092, A230093.

Programs

Haskell

a230094 n = a230094_list !! (n-1)
a230094_list = filter ((== 2) . a230093) [0..]
-- Reinhard Zumkeller, Oct 11 2013

Maple

For Maple code see A230093.

Mathematica

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 *)