A003052 -id:A003052 - cp's OEIS frontend

A342729 Self numbers in base i-1: numbers not of the form k + A066323(k).

1, 3, 5, 7, 9, 22, 24, 26, 39, 41, 43, 56, 58, 60, 73, 75, 77, 90, 92, 94, 107, 109, 111, 136, 138, 140, 153, 155, 157, 170, 172, 174, 199, 201, 203, 216, 218, 220, 233, 235, 237, 262, 264, 266, 279, 281, 283, 296, 298, 300, 313, 315, 317, 330, 332, 334, 347, 349

Offset: 1

Amiram Eldar, Mar 19 2021

Equivalently, self numbers in base -4, since A066323(k) is also the sum of the digits of k in base -4.
Analogous to self numbers (A003052) using base i-1 representation (A271472) instead of decimal expansion.
The number of terms not exceeding 10^k, for k=1,2,..., is 5, 20, 155, 1507, 15008, 150007, 1500014, 15000011. Is the asymptotic density of this sequence exactly 3/20?

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.

Cf. A007608, A066323, A066321, A271472, A342725, A342726, A342727, A342728.

Similar sequences: A003052 (decimal), A010061 (binary), A010064 (base 4), A010067 (base 6), A010070 (base 8), A339211 (Zeckendorf), A339212 (dual Zeckendorf), A339213 (base phi), A339214 (factorial base), A339215 (primorial base).

s[n_] := Module[{v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}}, Plus @@ Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; f[n_] := n + s[n]; m = 1000; Complement[Range[m], Select[Union@Array[f, m], # <= m &]]

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 *)

A171671 Square numbers not of form m + sum of digits of m.

Original entry on oeis.org

1, 9, 64, 121, 400, 5776, 6889, 7396, 8836, 9409, 10816, 12100, 17689, 18769, 27556, 29929, 30976, 33856, 34969, 37636, 49729, 65536, 69169, 69696, 70756, 75076, 75625, 76729, 80656, 110224, 124609, 126736, 132496, 134689, 156816, 162409

Offset: 1

Zak Seidov, Dec 15 2009

We may call these numbers the self or Colombian squares. Subsequence of A003052. There are 446 such self squares < 2*10^7, 218 odd and 228 even.

Kaprekar (1963) introduced these numbers and called them self-square numbers. - N. J. A. Sloane, Oct 30 2014

D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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]
Zak Seidov, Self squares.

Intersection of A000290 and A003052 (self or Colombian numbers).

Cf. A171672 (m^2 are self numbers), A062028 (a(n) = n + sum of the digits of n), A171673 (n and n^2 are self numbers), A382166.

A062028=Table[n+Total[IntegerDigits[n]],{n,0,20000000}];
se=Select[Complement[Range[0,20000000],A062028],IntegerQ[Sqrt[ # ]]&]

a(n) = A171672(n)^2. - Amiram Eldar, Mar 26 2025

Changed the word "safe" in this entry to "self". - N. J. A. Sloane, Feb 26 2017

A242403 Decimal expansion of the binary self-numbers density constant.

Original entry on oeis.org

2, 5, 2, 6, 6, 0, 2, 5, 9, 0, 0, 8, 8, 8, 2, 9, 2, 2, 1, 5, 5, 0, 6, 2, 7, 1, 4, 3, 2, 7, 8, 9, 4, 1, 4, 1, 8, 2, 5, 2, 1, 9, 3, 3, 9, 6, 2, 9, 7, 8, 4, 6, 1, 3, 0, 1, 6, 8, 6, 2, 1, 7, 2, 2, 9, 2, 2, 8, 0, 5, 4, 8, 4, 4, 7, 6, 6, 3, 2, 5, 6, 6, 9, 5, 9, 1, 4, 2, 4, 4, 7, 9, 3, 8, 6, 8, 8, 9, 4, 9

Offset: 0

Jean-François Alcover, May 13 2014

This constant is transcendental (Troi and Zannier, 1999). - Amiram Eldar, Nov 28 2020

			0.2526602590088829221550627143278941418252...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 179.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.
G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers, Bollettino dell'Unione Matematica Italiana, Serie 7, Vol. 9-A, No. 1 (1995), pp. 143-148.

G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers - II, Bollettino dell'Unione Matematica Italiana, Serie 8, Vol. 2-B, No. 2 (1999), pp. 397-399; alternative link.
Umberto Zannier, On the distribution of self-numbers, Proc. Amer. Math. Soc., Vol. 85, No. 1 (1982), pp. 10-14.
Index entries for transcendental numbers

Cf. A010061 (binary self numbers), A003052 (decimal self numbers), A010064, A010067, A010070, A092391, A228082.

m0 = 100; dm = 100; digits = 100; Clear[lambda]; lambda[m_] := lambda[m] = Total[1/2^Union[Table[n + Total[IntegerDigits[n, 2]], {n, 0, m}]]]^2/8 // N[#, 2*digits]& // RealDigits[#, 10, 2*digits]& // First; lambda[m0]; lambda[m = m0 + dm]; While[lambda[m] != lambda[m - dm], Print["m = ", m]; m = m + dm]; lambda[m][[1 ;; digits]]

Equals (1/8)*(Sum_{n not a binary self-number} 1/2^n)^2.

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

A350601 Array read by antidiagonals: row n lists even numbers that are "generated" (in Kaprekar's sense) in all bases 2, 4, 6, ..., 2n.

Original entry on oeis.org

0, 0, 2, 0, 2, 8, 0, 2, 10, 10, 0, 2, 10, 12, 12, 0, 2, 10, 14, 14, 14, 0, 2, 10, 14, 16, 16, 16, 0, 2, 10, 14, 22, 22, 22, 20, 0, 2, 10, 14, 22, 24, 24, 24, 22, 0, 2, 10, 14, 22, 24, 28, 28, 26, 24

Offset: 1

N. J. A. Sloane, Jan 08 2022

Max Alekseyev's PARI "Gen" program (see A010061) is essential for computing the rows. Cf. A349833.

			The initial rows of the array are:
  0, 2,  8, 10, 12, 14, 16, 20, 22, 24, 26, 28, 34, 36, 38, 40, 42, 44, 50, 52,  ... [the even terms of A228082]
  0, 2, 10, 12, 14, 16, 22, 24, 26, 28, 34, 36, 38, 40, 44, 50, 58, 60, 62, 66  ... [A349831]
  0, 2, 10, 14, 16, 22, 24, 28, 34, 36, 38, 44, 50, 58, 60, 62, 66, 68, 72, 74,  ... [A349832]
  0, 2, 10, 14, 22, 24, 28, 36, 38, 44, 50, 58, 60, 62, 66, 68, 74, 76, 82, 84,  ... [A349833]
  0, 2, 10, 14, 22, 24, 28, 36, 38, 44, 50, 58, 60, 62, 66, 68, 74, 76, 82, 84,  ...
  0, 2, 10, 14, 22, 28, 36, ...
  0, 2, 10, 14, 22, 36, ...
  0, 2, 10, 14, 22, 36,...
  0, 2, 10, 14, 22, ...
...
The rows converge to A230624, which is
  0, 2, 10, 14, 22, 38, 62, 94, 158, 206, 318, 382, 478, 606, 766, 958, 1022, ...
The initial antidiagonals are:
  0,
  0, 2,
  0, 2, 8,
  0, 2, 10, 10,
  0, 2, 10, 12, 12,
  0, 2, 10, 14, 14, 14,
  0, 2, 10, 14, 16, 16, 16,
  0, 2, 10, 14, 22, 22, 22, 20,
  0, 2, 10, 14, 22, 24, 24, 24, 22,
  0, 2, 10, 14, 22, 24, 28, 28, 26, 24,
  ...

Index entries for Colombian or self numbers and related sequences

The first few rows of the array are A228082 (even terms only), A349831, A349832, and A349833.

Cf. A003052, A010061, A230624.

[Needs checking and extending]

A003219 Self numbers divisible by sum of their digits (or, self numbers which are also Harshad numbers).

Original entry on oeis.org

1, 3, 5, 7, 9, 20, 42, 108, 110, 132, 198, 209, 222, 266, 288, 312, 378, 400, 468, 512, 558, 648, 738, 782, 804, 828, 918, 1032, 1098, 1122, 1188, 1212, 1278, 1300, 1368, 1458, 1526, 1548, 1638, 1704, 1728, 1818, 1974, 2007, 2022, 2088, 2112, 2156, 2178

Offset: 1

N. J. A. Sloane

D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
D. R. Kaprekar, The Mathematics of the New Self Numbers (Part V). 311 Devlali Camp, Devlali, India, 1967.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..7045

Cf. A003052, A005349.

Select[list=Range[2178]; Complement[list,Table[n+Total[IntegerDigits[n]],{n,list}]], IntegerQ[#/Total[IntegerDigits[#]]] &] (* Jayanta Basu, May 05 2013 *)

More terms from James Sellers, Jul 06 2000

A163128 a(n) is the n-th self-number minus n.

Original entry on oeis.org

0, 1, 2, 3, 4, 14, 24, 34, 44, 54, 64, 74, 84, 94, 95, 105, 115, 125, 135, 145, 155, 165, 175, 185, 186, 196, 206, 216, 226, 236, 246, 256, 266, 276, 277, 287, 297, 307, 317, 327, 337, 347, 357, 367, 368, 378, 388, 398, 408, 418, 428, 438, 448, 458, 459, 469, 479, 489, 499

Offset: 1

Juri-Stepan Gerasimov, Jul 21 2009

			a(6) = 20 - 6 = 14.
a(7) = 31 - 7 = 24.

Cf. A000027, A003052.

A007953 := proc(n) add(d,d=convert(n,base,10)) ; end:
isA003052 := proc(n) for k from 1 to n do if k+A007953(k) = n then RETURN(false) ; fi; od: true; end:
A003052 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA003052(a) then RETURN(a) ; fi; od: fi; end:
A163128 := proc(n) A003052(n)-n ; end:
for n from 1 to 100 do printf("%d,",A163128(n)) ; od: # R. J. Mathar, Jul 31 2009

a(n) = A003052(n) - n.

Entries checked by R. J. Mathar, Jul 31 2009

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.

A336984 Colombian numbers that are also Bogotá numbers.

Original entry on oeis.org

1, 9, 42, 64, 75, 255, 312, 378, 525, 648, 738, 1111, 1278, 2224, 2448, 2784, 2817, 3504, 3864, 3875, 4977, 5238, 5495, 5888, 8992, 9712, 10368, 11358, 11817, 12348, 12875, 13136, 13584, 13775, 13832, 13944, 15351, 15384, 15744, 15900, 16912, 17768, 18095, 19344, 20448

Offset: 1

Bernard Schott, Aug 22 2020

Equivalently, numbers m that are not of the form k + sum of digits of k for any k (A003052), but are of the form q * product of digits of q for some q (A336826).
Repunits are trivially Bogotá numbers but the indices m of the repunits R_m that are Colombian numbers are in A337208. No known prime belongs to this sequence (see A004023).
A336983, A336985, A336986 and this sequence form a partition of the set of positive integers N*.

			42 = 21 * (2*1) is a Bogotá number and there does not exist m < 42 such that 42 = m + sum of digits of m, hence 42 is a Colombian number and 42 is a term.
56 = 14 * (1*4) is a Bogotá number but as 56 = 46 + (4+6), 56 is not a Colombian number, hence 56 is not a term.
648 = 36 * (3*6) = 81 * (8*1) but there does not exist m < 648 such that 648 = m + sum of digits of m, hence 648 is a Colombian number, so 648 is a term that has two different representations as the product of a number and of its decimal digits.

David A. Corneth, Table of n, a(n) for n = 1..10000
Giovanni Resta, Self or Colombian number, Numbers Aplenty.
Puzzling Stackexchange, Pairs of Bogotá Numbers.
Index to sequences related to Colombian numbers.

Intersection of A003052 and A336826.
Cf. A336983 (Bogotá and not Colombian), A336985 (Colombian not Bogotá), A336986 (not Colombian and not Bogotá).
Cf. A336944, A336983, A337208.

m = 21000; Intersection[Complement[Range[m], 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 22 2020 *)

lista(nn) = Vec(setintersect(setminus([1..nn], Set(vector(nn, k, k+sumdigits(k)))), Set(vector(nn, k, k*vecprod(digits(k)))))); \\ Michel Marcus, Aug 23 2020

cp's OEIS Frontend

A342729 Self numbers in base i-1: numbers not of the form k + A066323(k).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

A171671 Square numbers not of form m + sum of digits of m.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A242403 Decimal expansion of the binary self-numbers density constant.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A336983 Bogota numbers that are not Colombian numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A350601 Array read by antidiagonals: row n lists even numbers that are "generated" (in Kaprekar's sense) in all bases 2, 4, 6, ..., 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A003219 Self numbers divisible by sum of their digits (or, self numbers which are also Harshad numbers).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Extensions

A163128 a(n) is the n-th self-number minus n.

Views