

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

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

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).
Santanu Bandyopadhyay, SelfNumber, Indian Institute of Technology Bombay (Mumbai, India, 2020).
Santanu Bandyopadhyay, SelfNumber, Indian Institute of Technology Bombay (Mumbai, India, 2020). [Local copy]


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



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..]


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



