A230094 -id:A230094 - cp's OEIS frontend

A230093 Number of values of k such that k + (sum of digits of k) is n.

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1

Offset: 0

N. J. A. Sloane, Oct 10 2013

a(n) is the number of times n occurs in A062028.

For n>=1, a(10^n) = a(9*n-1). - Max Alekseyev, Feb 23 2021

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A006064, A007953 (sum of digits), A062028 (n + sum of its digits), A004207, A228085, A003052, A176995, A225793, A230094, A055642.

Cf. A107740 (this applied to primes).

a230093 n = length $ filter ((== n) . a062028) [n - 9 * a055642 n .. n]  -- Reinhard Zumkeller, Oct 11 2013

# Maple code for A062028, A230093, A003052, A225793, A230094.
with(LinearAlgebra):
read transforms; # to get digsum
M := 1000; A062028 := Array(0..M); A230093 := Array(0..M);
for n from 0 to M do
   m := n+digsum(n);
   A062028[n] := m;
   if m <= M then A230093[m] := A230093[m]+1; fi;
od:
t1:=[seq(A062028[i],i=0..M)]; # A062028 as list (but incorrect offset 1)
t2:=[seq(A230093[i],i=0..M)]; # A230093 as list, but then a(0) has index 1
# A003052 := COMPl(t1); # COMPl has issues, may be incorrect for M <> 1000
ctmax:=4;
for h from 0 to ctmax do ct[h] := []; od:
for i from 1 to M do
   h := lis2[i];
   if h <= ctmax then ct[h] := [op(ct[h]),i]; fi;
od:
A225793 := ct[1]; A230094 := ct[2]; # A003052 := ct[0]; # see there for better code

Module[{nn=110,a,b,c,d},a=Tally[Table[x+Total[IntegerDigits[x]],{x,0,nn}]];b=a[[All,1]];c={#,0}&/@Complement[Range[nn],b];d=Sort[Join[a,c]];d[[All, 2]]] (* Harvey P. Dale, Jun 12 2019 *)

apply( A230093(n)=sum(i=n>0,min(9*logint(n+!n,10)+8,n\2),sumdigits(n-i)==i), [1..150]) \\ M. F. Hasler, Nov 08 2018

Edited by M. F. Hasler, Nov 08 2018

A225793 Numbers n that can be uniquely expressed as (m + sum of digits of m) for some m.

Original entry on oeis.org

2, 4, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77

Offset: 1

Jayanta Basu, Jul 27 2013

Subset of A176995; first member in A176995 that is not here is 101, next is 103 (cf. A230094).

A230093(a(n)) = 1. - Reinhard Zumkeller, Oct 11 2013

			100 is a member as 100 = 86 + sum of digits of (86). 101 is not a member since both 91 and 100 generate 101. Again 103 is not a member as 92 and 101 generate 103.

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)
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)

Jayanta Basu and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from Jayanta Basu)
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A062028, A004207, A230093, A003052, A176995, A230094.

a225793 n = a225793_list !! (n-1)
a225793_list = filter ((== 1) . a230093) [1..]
-- Reinhard Zumkeller, Oct 11 2013

For Maple code see A230093. - N. J. A. Sloane, Oct 11 2013

co[n_] := Count[Range[n - 1], _?(# + Total[IntegerDigits[#]] == n &)]; Select[Range[100], co[#] == 1 &]
Select[Tally[Table[m+Total[IntegerDigits[m]],{m,100}]],#[[2]]==1&][[All, 1]]// Sort (* Harvey P. Dale, Aug 23 2017 *)

A336983 Bogota numbers that are not Colombian numbers.

Original entry on oeis.org

4, 11, 16, 24, 25, 36, 39, 49, 56, 81, 88, 93, 96, 111, 119, 138, 144, 164, 171, 192, 224, 242, 250, 297, 336, 339, 366, 393, 408, 422, 448, 456, 488, 497, 516, 520, 522, 564, 575, 696, 704, 744, 755, 777, 792, 795, 819, 848, 884, 900, 912, 933, 944, 966, 992

Offset: 1

Bernard Schott, Aug 10 2020

Equivalently, numbers m that are of the form k + sum of digits of k for some k (A176995), and also of the form q * product of digits of q for some q (A336826).
Repunits are trivially Bogota numbers but the indices m of the repunits R_m that are not Colombian numbers are in A337139; also, all known repunit primes are terms (A004023) [see examples for primes R_2, R_19 and R_23].
35424 is the smallest term that belongs to both A230094 and A336944 (see last example).

			R_2 = 11 = 10 + (1+0) = 11 * (1*1) is a term;
24 = 21 + (2+1) = 12 * (1*2) is a term;
39 = 33 + (3+3) = 13 * (1*3) is a term;
R_19 = 1111111111111111079 + (16*1+7+9) = 1111111111111111111 * (1^19) hence R_19 is a term;
R_23 = 11111111111111111111077 + (20*1+7+7) = 11111111111111111111111 * (1^23) hence R_23 is a term;
42 = 21 * (2*1) is a Bogota number but there does not exist m < 42 such that 42 = m + sum of digits of m, hence 42 that is also a Colombian number is not a term.
35424 = 35406 + (3+5+4+0+6) = 35397 + (3+5+3+9+7) = 2214 * (2*2*1*4) = 492 * (4*9*2).

Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Puzzling Stackexchange, Pairs of Bogotá Numbers.
Index to sequences related to Colombian numbers.

Intersection of A176995 and A336826.
Cf. A003052 (Colombian), A336984 (Bogota and Colombian), A336985 (Colombian not Bogota), A336986 (not Colombian and not Bogota).
Cf. A230093, A230094, A336944, A337139.

m = 1000; Intersection[Select[Union[Table[n + Plus @@ IntegerDigits[n], {n, 1, m}]], # <= m &], Select[Union[Table[n * Times @@ IntegerDigits[n], {n, 1, m}]], # <= m &]] (* Amiram Eldar, Aug 10 2020 *)

lista(nn) = Vec(setintersect(Set(vector(nn, k, k+sumdigits(k))), Set(vector(nn, k, k*vecprod(digits(k)))))); \\ Michel Marcus, Aug 20 2020

A230100 Numbers that can be expressed as (m + sum of digits of m) in exactly three ways.

Original entry on oeis.org

10000000000001, 10000000000003, 10000000000005, 10000000000007, 10000000000009, 10000000000011, 10000000000013, 10000000000015, 10000000000102, 10000000000104, 10000000000106, 10000000000108, 10000000000110, 10000000000112, 10000000000114, 10000000000116

Offset: 1

N. J. A. Sloane, Oct 12 2013 - Oct 25 2013

Let f(n) = n + (sum of digits of n) = A062028(n).
Let g(m) = number of n such that f(n) = m (i.e. the number of inverses of m), A230093(m).
Numbers m with g(m) = 0 are called the Self or Colombian numbers, A003052.
Numbers m with g(m) = 1 give A225793.
Numbers m with g(m) = 2 give A230094.
The present sequence gives numbers m such that A230093(m) = 3.
The smallest term, a(1) = 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(2) onwards were computed by Donovan Johnson, Oct 12 2013, and on Oct 20 2013 he completed a search of all numbers below 10^13 and verified that 10^13 + 1 is indeed the smallest term.
See A006064 for much more about this question.
Numbers m with g(m) = 4 give A377422. - Daniel Mondot, Oct 29 2024

			There are exactly three numbers, 9999999999892, 9999999999901 and 10000000000000, whose image under n->f(n) is 10000000000001, so 10^13+1 is a member of the sequence.

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

Daniel Mondot, 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.
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
Index entries for Colombian or self numbers and related sequences

Cf. A006064, A062028, A230093.

A377422 Numbers that can be expressed as (m + sum of digits of m) in exactly four ways.

Original entry on oeis.org

1000000000000000000000102, 1000000000000000000000104, 1000000000000000000000106, 1000000000000000000000108, 1000000000000000000000110, 1000000000000000000000112, 1000000000000000000000114, 2000000000000000000000103, 2000000000000000000000105, 2000000000000000000000107, 2000000000000000000000109

Offset: 1

Daniel Mondot, Oct 27 2024

Numbers k such that A230093(k) = 0 give A003052, the Self or Colombian numbers.
Numbers k such that A230093(k) = 1 give A225793.
Numbers k such that A230093(k) = 2 give A230094.
Numbers k such that A230093(k) = 3 give A230100.
Numbers k such that A230093(k) = 4 give this sequence.

			There are exactly four numbers, 999999999999999999999894, 999999999999999999999903, 1000000000000000000000092, and 1000000000000000000000101, whose image under n->f(n) is 1000000000000000000000104, so 1000000000000000000000104 is a member of the sequence.

Daniel Mondot, 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.

Cf. A230100, A006064, A062028, A230093.

Corrected by Daniel Mondot, Apr 17 2025

A377423 Distinct values of the number of integers between consecutive self numbers (A163139), in order of occurrence.

Original entry on oeis.org

1, 10, 14, 27, 40, 53, 66, 79, 92, 105, 118, 100, 117, 130, 143, 156, 169, 182, 195, 208, 23, 89, 203, 220, 233, 246, 259, 272, 285, 298, 34, 78, 293, 306, 323, 336, 349, 362, 375, 388, 45, 67, 383, 396, 409, 426, 439, 452, 465, 478, 56, 473, 486, 499

Offset: 1

Daniel Mondot, Oct 27 2024

Each new value is typically found between self numbers located around 10^k, for some k.

This sequences exhibits interesting patterns, for instance, many new numbers are 13 apart.

			Between the first 2 self numbers 1 and 3, there is 1 integer. So 1 is in the sequence
The next new gap is between 9 and 20, with 10 integers, so 10 is in the sequence.
The next new gap is between 1006 and 1021, with 14 integers, so 14 is in the sequence.

Daniel Mondot, Table of n, a(n) for n = 1..10000
Daniel Mondot, First self number found after a new gap value

Cf. A003052, A163128, A163139, A225793, A230093, A230094, A377422, A377472, A377473.

a(n) = A377473(n)-1. - Daniel Mondot, Apr 17 2025

A230304 a(n) = 10^( (10^n-1)/9 + n) + 1.

Original entry on oeis.org

101, 10000000000001, 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

Offset: 1

N. J. A. Sloane, Oct 26 2013

Makowski observes that A230093(a(n)) >= 2 for all n >= 1.

Makowski, Andrzej. On Kaprekar's "junction numbers''. Math. Student 34 1966 77 (1967). MR0223292 (36 #6340)

Index entries for Colombian or self numbers and related sequences

Cf. A006064, A230093, A230094, A230100.

cp's OEIS Frontend

A230093 Number of values of k such that k + (sum of digits of k) is n.

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A225793 Numbers n that can be uniquely expressed as (m + sum of digits of m) for some m.

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A336983 Bogota numbers that are not Colombian numbers.

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A230100 Numbers that can be expressed as (m + sum of digits of m) in exactly three ways.

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A377422 Numbers that can be expressed as (m + sum of digits of m) in exactly four ways.

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A377423 Distinct values of the number of integers between consecutive self numbers (A163139), in order of occurrence.

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A230304 a(n) = 10^( (10^n-1)/9 + n) + 1.

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