

A230094


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


6



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
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OFFSET

1,1


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 socalled "junction numbers") would produce a sequence which agrees with A230094 up to 10^13.
Makowski shows that the sequence of junction numbers is infinite.


REFERENCES

Max A. Alekseyev, Donovan Johnson and N. J. A. Sloane, On Kaprekar's Junction Numbers, in preparation, 2017.
Joshi, V. S. A note on selfnumbers. Volume dedicated to the memory of V. Ramaswami Aiyar. Math. Student 39 (1971), 327328 (1972). MR0330032 (48 #8371)
D. R. Kaprekar, Puzzles of the SelfNumbers. 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 7984 (1967). MR0229573 (37 #5147)


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Santanu Bandyopadhyay, SelfNumber, Indian Institute of Technology Bombay (Mumbai, India, 2020).
David A. Corneth, Examples
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
Index entries for Colombian or self numbers and related sequences


EXAMPLE

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).


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 *)


PROG

(Haskell)
a230094 n = a230094_list !! (n1)
a230094_list = filter ((== 2) . a230093) [0..]
 Reinhard Zumkeller, Oct 11 2013


CROSSREFS

Cf. A003052, A007953, A004207, A048528, A062028, A176995, A225793, A227915, A230092, A230093.
Sequence in context: A271642 A164849 A162671 * A030474 A162199 A195469
Adjacent sequences: A230091 A230092 A230093 * A230095 A230096 A230097


KEYWORD

nonn,base


AUTHOR

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


STATUS

approved



