A006064 -id:A006064 - cp's OEIS frontend

A238843 Smallest number m such that u + (sum of base-9 digits of u) = m has exactly n solutions.

0, 10, 82, 740, 282429536492, 109418989131512359292, 193632597890512706847971583764083347958511186984324587565465147107798425867049291402906445603076812

Offset: 1

N. J. A. Sloane, Mar 19 2014

Max Alekseyev, Table of expressions for a(n), for n=1..100
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.

Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

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

A358352 a(n) is the smallest number k such that A358351(k) = n.

Original entry on oeis.org

1, 3, 26, 38, 380, 1116, 12912, 95131, 342038, 3320210, 494204209, 773089018

Offset: 0

Bernard Schott, Nov 19 2022

			19+sod(19)+pod(19) = 24+sod(24)+pod(24) = 31+sod(31)+pod(31) = 38, and there is no integer < 38 for which function A161351 has 3 preimages, so a(3) = 38.

Rémy Sigrist, C program

Cf. A006064, A007953, A007954, A161351, A358350, A358351.

C
```
See Links section.
```

f[n_] := n + Total[(d = IntegerDigits[n])] + Times @@ d; s = With[{m = 10^7}, BinCounts[Table[f[n], {n, 1, m}], {1, m, 1}]]; FirstPosition[s, #] & /@ Range[0, Max[s]] // Flatten (* Amiram Eldar, Nov 19 2022 *)

first(n) = my(res = vector(n)); for(i = 1, n, c = i + sumdigits(i) + vecprod(digits(i)); if(c <= n, res[c]++ ) ); res; \\ A358351
lista(nn) = my(v=first(nn)); for (n=0, 20, my(vs = select(x->(x==n), v, 1)); if (#vs, print1(vs[1], ", "), break);); \\ Michel Marcus, Nov 20 2022

a(4)-a(5) from Michel Marcus, Nov 19 2022
a(6)-a(9) from Amiram Eldar, Nov 19 2022
a(10)-a(11) from Rémy Sigrist, Nov 20 2022

A239896 Generalized ternary Thue-Morse sequence arising from junction numbers problem in base 10.

Original entry on oeis.org

0, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3, 8, 3

Offset: 0

Max Alekseyev and N. J. A. Sloane, Apr 05 2014

nonn

See Section 9 in the linked paper.

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. A006064, A010060, A239110.

a(n) = A006064(n) mod 9.

A320870 Irregular table: row n >= 0 lists numbers m >= 0 such that n = A062028(m) := m + sum of digits of m.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 6, 11, 7, 12, 8, 13, 9, 14, 15, 20, 16, 21, 17, 22, 18, 23, 19, 24, 25, 30, 26, 31, 27, 32, 28, 33, 29, 34, 35, 40, 36, 41, 37, 42, 38, 43, 39, 44, 45, 50, 46, 51, 47, 52, 48, 53, 49, 54, 55, 60, 56, 61, 57, 62, 58, 63, 59, 64, 65, 70, 66, 71, 67, 72, 68, 73, 69, 74, 75, 80, 76, 81, 77, 82, 78, 83, 79, 84, 85, 90

Offset: 0

M. F. Hasler, Nov 09 2018

Row lengths are given by A230093.

			The first nonempty rows are:
    n  | list of m
    0  | 0        // since 0 = 0 + 0
    2  | 1        // since 2 = 1 + 1
    4  | 2        // etc.
    6  | 3        // Below 10 every odd row is empty, but thereafter,
    8  | 4        // only rows 20, 31, 42, ..., 108 (steps of 11),
   10  | 5        // 110, 121, 132, ..., 198, etc. are empty.
   11  | 10       // Since 11 = 10 + (1 + 0)
   12  | 6
   13  | 11       // The first prime that yields a prime: 11 + (1 + 1) = 13.
     (...)
  100  | 86       // The first row of length 2 is 101:
  101  | 91, 100  // 101 = 91 + (9 + 1) = 100 + (1 + 0 + 0)
  102  | 87
     (...)

Robert Israel, Table of n, a(n) for n = 0..9971 (rows 0 to 10000, flattened)

Cf. A007953 (sum of digits of n), A062028 (n + digit sum of n).
Cf. A230093 (number of m such that m + (sum of digits of m) is n).
Cf. A006064 (least m with row length n),
Cf. A003052 (Self or Colombian numbers: rows of length 0), A006378 (Colombian primes).
Cf. A320881 (indices of rows containing a prime), A048520 (primes among these).

N:= 100: # for rows 0 to N, flattened
for i from 0 to N do V[i]:= NULL od:
for i from 0 to N-1 do
  v:= convert(convert(i,base,10),`+`);
  if v <= N then V[v]:= V[v],i fi
od:
seq(V[i],i=1..N); # Robert Israel, Jul 21 2025

A320870_row(n)=if(n,select(m->m+sumdigits(m)==n,[max(n-9*logint(n,10)+8,n\/2)..n-1]),[0])

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.

A309393 Let f(n) be equal to n + S(n) + S(S(n)) ... + S(S(S..(n))), where the last term is less than 10 and S(n) is the sum of digits. This is the sequence of numbers k such that the equation f(x) = k has a record number of solutions.

Original entry on oeis.org

1, 30, 66, 204, 819, 70032, 3000000000096

Offset: 1

Daniel Starodubtsev, Jul 28 2019

Conjecture from Daniel Starodubtsev and Dmitry Petukhov, Nov 19 2019: (Start)
a(8) = 20000000000000046;
a(9) = 8900000000000000000000127. (End)

			a(4) = 204, because 204 = f(179) = f(185) = f(191) = f(201), which has more solutions than any smaller number.

Cf. A007953, A006064.

T = 0*Range[10^5]; f[n_] := Block[{x=n, s=n}, While[x >= 10, x = Plus@@ IntegerDigits[x]; s += x]; s]; Do[v = f[i]; If[v <= 10^5, T[[v]]++], {i, 10^5}]; Flatten[Position[T, #, 1, 1] & /@ Range[6]] (* Giovanni Resta, Jul 30 2019 *)

f(n) = {s=n;m=n;while(sumdigits(s)>9,s=sumdigits(s);m+=s);if(n<10,m=0);m+sumdigits(s);}
g(n) = sum(k=1,n,f(k)==n);
lista(NN) = {x=1;print1(1);for(n=2,NN,if(g(n)>x,x=g(n);print1(", ",n)))} \\ Jinyuan Wang, Jul 31 2019

a(7) from Bert Dobbelaere, Aug 15 2019

cp's OEIS Frontend

A238843 Smallest number m such that u + (sum of base-9 digits of u) = m has exactly n solutions.

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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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A358352 a(n) is the smallest number k such that A358351(k) = n.

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C

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A239896 Generalized ternary Thue-Morse sequence arising from junction numbers problem in base 10.

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A320870 Irregular table: row n >= 0 lists numbers m >= 0 such that n = A062028(m) := m + sum of digits of m.

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Maple

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

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A309393 Let f(n) be equal to n + S(n) + S(S(n)) ... + S(S(S..(n))), where the last term is less than 10 and S(n) is the sum of digits. This is the sequence of numbers k such that the equation f(x) = k has a record number of solutions.

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Author

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Comments

Examples

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Programs

Mathematica

PARI

Extensions