A176995 -id:A176995 - cp's OEIS frontend

A358512 a(n) is the smallest number k with exactly n divisors that can be written in the form m + digsum(m), for some m (A176995).

1, 2, 4, 8, 12, 30, 24, 80, 48, 72, 96, 192, 120, 180, 288, 612, 240, 624, 420, 360, 480, 900, 1632, 960, 1200, 720, 840, 1560, 2100, 1260, 1440, 3420, 2640, 3024, 1680, 2880, 8316, 4620, 3600, 3780, 4200, 2520, 3360, 6240, 9900, 6300, 7200, 8640, 6720, 13200, 7920

Offset: 0

Marius A. Burtea, Dec 04 2022

			1 cannot be written in the form m + digsum(m), so a(0) = 1.
2 has divisors 1 and 2, and only 2 is written 2 = 1 + digsum(1), so a(1) = 2.
3 has divisors 1 and 3 that cannot be written in the form m + digsum(m).
4 has divisors 1, 2, 4, but only 2 = 1 + digsum(1) and 4 = 2 + digsum(2), so a(2) = 4.

David A. Corneth, Table of n, a(n) for n = 0..2499

Cf. A003052, A062028, A176995, A230093.

f:=func; a:=[]; for n in [0..50] do k:=1; while #[d:d in Divisors(k)|f(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

is_A003052(n)={for(i=1, min(n\2, 9*#digits(n)), sumdigits(n-i)==i && return); n}
a(n) = my(k=1); while (sumdiv(k, d, !is_A003052(d)) != n, k++); k; \\ Michel Marcus, Dec 13 2022

A003052 Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).

Original entry on oeis.org

1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, 501, 512, 514, 525

Offset: 1

N. J. A. Sloane

From Amiram Eldar, Nov 28 2020: (Start)
The term "self numbers" was coined by Kaprekar (1959). The term "Colombian number" was coined by Recamán (1973) of Bogota, Colombia.
The asymptotic density of this sequence is approximately 0.0977778 (Guaraldo, 1978). (End)

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24.
Martin Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
V. S. Joshi, A note on self-numbers. Volume dedicated to the memory of V. Ramaswami Aiyar. Math. Student, Vol. 39 (1971), pp. 327-328. MR0330032 (48 #8371).
D. R. Kaprekar, Puzzles of the Self-Numbers. 311 Devlali Camp, Devlali, India, 1959.
D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately Printed, 311 Devlali Camp, Devlali, India, 1963.
D. R. Kaprekar, The Mathematics of the New Self Numbers (Part V). 311 Devlali Camp, Devlali, India, 1967.
Bernardo Recamán, The Bogota Puzzles, Dover Publications, Inc., 2020, chapter 36, p. 33.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 384-386.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Author?, J. Recreational Math., vol. 23, no. 1, p. 244, 1991.

Reinhard Zumkeller, 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.
Christian N. K. Anderson, Ulam Spiral of the first 5000 self numbers.
Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020).
Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020). [Local copy]
Martin Gardner and N. J. A. Sloane, Correspondence, 1973-74
Rosalind Guaraldo, On the Density of the Image Sets of Certain Arithmetic Functions - II, The Fibonacci Quarterly, Vol. 16, No. 5 (1978), pp. 481-488.
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, 1963. [annotated and scanned]
Andrzej Makowski, On Kaprekar's "junction numbers", Math. Student, Vol. 34, No. 2 (1966), p. 77. MR0223292 (36 #6340); entire issue.
A. Narasinga Rao, On a technique for obtaining numbers with a multiplicity of generators, Math. Student, Vol. 34, No. 2 (1966), pp. 79-84. MR0229573 (37 #5147); entire issue.
Bernardo Recamán, Problem E2408, Amer. Math. Monthly, Vol. 80, No. 4 (1973), p. 434; Colombian Numbers, solution to Problem E2408 by D. W. Bange, ibid., Vol. 81, No. 4 (1974), p. 407.
Giovanni Resta, Self or Colombian numbers, Numbersaplenty, 2013.
Richard Schorn, Kaprekar's Sequence and his "Selfnumbers", DERIVE Newsletter, #53 (2004), pp. 30-32.
Walter Schneider, Self Numbers, 2000-2003.
Walter Schneider, Self Numbers, 2000-2003 (unpublished; local copy)
N. J. A. Sloane, Martin Gardner and D. R. Kaprekar, Correspondence, 1974 [Scanned letters]
Terry Trotter, Charlene Numbers
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.
U. Zannier, On the distribution of self-numbers, Proc. Amer. Math. Soc., Vol. 85, No. 1 (1982), pp. 10-14.
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A232229, A062028, A055642, A282711.
For self primes, i.e., self numbers which are primes, see A006378.
Complement of A176995.
See A010061 for the binary version, A283002 for a base-100 version.
Cf. A247104 (subsequence of squarefree terms).
Cf. A377472 for first differences, A377474 for indices where new differences appear.

a003052 n = a003052_list !! (n-1)
a003052_list = filter ((== 0) . a230093) [1..]
-- Reinhard Zumkeller, Oct 11 2013, Aug 21 2011

isA003052 := proc(n) local k ; for k from 0 to n do if k+A007953(k) = n then RETURN(false): fi; od: RETURN(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: # R. J. Mathar, Jul 27 2009

nn = 525; Complement[Range[nn], Union[Table[n + Total[IntegerDigits[n]], {n, nn}]]] (* T. D. Noe, Mar 31 2013 *)

is_A003052(n)={for(i=1,min(n\2,9*#digits(n)), sumdigits(n-i)==i && return); n}  \\ M. F. Hasler, Mar 20 2011, updated Nov 08 2018

is(n) = {if(n < 30, return((n < 10 && n%2 == 1) || n == 20)); qd = 1 + logint(n, 10); r = 1 + (n-1)%9; h = (r + 9 * (r%2))/2; ld = 10; while(h + 9*qd >= n % ld, ld*=10); vs = vecsum(digits(n \ ld)); n %= ld; for(i = 0, qd, if(vs + vecsum(digits(n - h - 9*i)) == h + 9*i, return(0))); 1} \\ David A. Corneth, Aug 20 2020

A230093(a(n)) = 0. - Reinhard Zumkeller, Oct 11 2013

In fact this defines the sequence: x is in the sequence iff A230093(x) = 0. - M. F. Hasler, Nov 08 2018

More terms from James Sellers, Jul 06 2000

A062028 a(n) = n + sum of the digits of n.

Original entry on oeis.org

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

Offset: 0

Amarnath Murthy, Jun 02 2001

a(n) = A248110(n,A007953(n)). - Reinhard Zumkeller, Oct 01 2014

			a(34) = 34 + 3 + 4 = 41, a(40) = 40 + 4 = 44.

Harry J. Smith and N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (first 1000 terms were computed by Harry J. Smith)
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A006378, A048521, A107740, A066568, A064806, A176995 (range), A003052, A230093, A248110.
Indices of: A047791 (primes), A107743 (composites), A066564 (squares), A084661 (cubes).
Iterations: A004207 (start=1), A016052 (start=3), A007618 (start=5), A006507 (start=7), A016096 (start=9).

a062028 n = a007953 n + n  -- Reinhard Zumkeller, Oct 11 2013

with(numtheory): for n from 1 to 100 do a := convert(n,base,10):
c := add(a[i],i=1..nops(a)): printf(`%d,`,n+c); od:
A062028 := n -> n+add(i,i=convert(n,base,10)) # M. F. Hasler, Nov 08 2018

Table[n + Total[IntegerDigits[n]], {n, 0, 100}]

A062028(n)=n+sumdigits(n) \\ M. F. Hasler, Jul 19 2015

def a(n): return n + sum(map(int, str(n)))
print([a(n) for n in range(71)]) # Michael S. Branicky, Jan 09 2023

a(n) = n + A007953(n).

a(n) = A160939(n+1) - 1. - Filip Zaludek, Oct 26 2016

More terms from Vladeta Jovovic, Jun 05 2001

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

Original entry on oeis.org

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

A230099 a(n) = n + (product of digits of n).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 60, 67, 74, 81, 88, 95, 102, 109, 116, 123, 70, 78, 86, 94, 102, 110, 118, 126

Offset: 0

N. J. A. Sloane, Oct 12 2013

A230099, A063114, A098736, A230101 are analogs of A092391 and A062028.

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

Cf. A007954, A011540, A092391, A062028, A004207, A230093, A003052, A176995.

Cf. also A063114, A098736, A230101, A230103, A230104, A230105, A062028, A230102, A063108.

a230099 n = a007954 n + n  -- Reinhard Zumkeller, Oct 13 2013

with transforms; [seq(n+digprod(n), n=0..200)];

a(n) = if (n, n + vecprod(digits(n)), 0); \\ Michel Marcus, Dec 18 2018

from math import prod
def a(n): return n + prod(map(int, str(n)))
print([a(n) for n in range(78)]) # Michael S. Branicky, Jan 09 2023

a(n) = n iff n contains a digit 0 (A011540). - Bernard Schott, Jul 31 2023

A230094 Numbers that can be expressed as (m + sum of digits of m) in exactly two ways.

Original entry on oeis.org

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

Offset: 1

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

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

			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).

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)
D. R. Kaprekar, Puzzles of the Self-Numbers. 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 79--84 (1967). MR0229573 (37 #5147)

Reinhard Zumkeller, 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.
Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020).
Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020). [Local copy]
David A. Corneth, Examples
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]
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A007953, A004207, A048528, A062028, A176995, A225793, A227915, A230092, A230093.

a230094 n = a230094_list !! (n-1)
a230094_list = filter ((== 2) . a230093) [0..]
-- Reinhard Zumkeller, Oct 11 2013

Maple
```
For Maple code see A230093.
```

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 *)

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 *)

A337718 Numbers that can be written as (m + product of digits of m) for some m.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 38, 40, 41, 42, 44, 45, 46, 47, 50, 54, 55, 56, 58, 60, 62, 65, 66, 67, 68, 70, 74, 75, 78, 80, 81, 85, 86, 88, 89, 90, 92, 94, 95, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109

Offset: 1

Bernard Schott, Sep 16 2020

Every integer that contains a digit 0 is a term (A011540).
When R_m with m >= 1 is in A002275, then R_m + 1 is a term (A047855 \ {1}).
Near similar:
-> Not-Colombian (A176995) are numbers that can be written as (m + sum of digits of m) for some m.
-> Bogotá numbers (A336826) are numbers that can be written as (m * product of digits of m) for some m.

			10 = 5 + 5 = 10 + (1*0) and 22 = 16 + (1*6) are terms.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Subsequences: A011540, A047855 \ {1}.
Range of A230099.
Cf. A176995 (not Colombian), A336826 (Bogotá numbers).

m = 100; Select[Union[Table[n + Times @@ IntegerDigits[n], {n, 0, m}]], # <= m &] (* Amiram Eldar, Sep 16 2020 *)

isok(m) = {if (m==0, return (1)); for (k=1, m,  if (k+vecprod(digits(k)) == m, return (1)););} \\ Michel Marcus, Sep 17 2020

from math import prod
def b(n): return n + prod(map(int, str(n)))
def aupto(n): return sorted(set(b(m) for m in range(n+1) if b(m) <= n))
print(aupto(109)) # Michael S. Branicky, Jan 09 2023

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

A358350 Numbers that can be written as (m + sum of digits of m + product of digits of m) for some m.

Original entry on oeis.org

3, 6, 9, 11, 12, 14, 15, 17, 18, 20, 21, 22, 23, 24, 26, 27, 29, 30, 32, 33, 34, 35, 38, 42, 43, 44, 46, 48, 50, 53, 54, 55, 56, 58, 62, 63, 66, 68, 69, 73, 74, 76, 77, 78, 80, 82, 83, 86, 88, 90, 92, 95, 97, 98, 99, 101, 103, 104, 105, 106, 107, 108, 109, 110

Offset: 1

Bernard Schott, Nov 11 2022

Integers that are in A161351.
(i) Can arbitrarily long sets of consecutive integers be found in this sequence?
(ii) Is the gap between two consecutive terms bounded?
A000533 \ {1} is a subsequence.
This has the same asymptotic density, approximately 0.9022222, as A176995, since the asymptotic density of non-pandigital numbers is 0. - Charles R Greathouse IV, Nov 16 2022

			A161351(23) = 23 + (2+3) + (2*3) = 34 so 34 is a term.
There is no integer du_10 such that du + (d+u) + (d*u) = 31, so 31 is not a term.

David A. Corneth, Table of n, a(n) for n = 1..10000

Range of A161351.
Similar: A176995 (m+digitsum), A336826 (m*digitprod), A337718 (m+digitprod).
Cf. A000533.

f[n_] := n + Total[(d = IntegerDigits[n])] + Times @@ d; With[{m = 110}, Select[Union[Table[f[n], {n, 1, m}]], # <= m &]] (* Amiram Eldar, Nov 11 2022 *)

f(n) = my(d=digits(n)); n + vecsum(d) + vecprod(d); \\ A161351
lista(nn) = select(x->(x<=nn), Set(vector(nn, k, f(k)))); \\ Michel Marcus, Nov 12 2022

from math import prod
def sp(n): d = list(map(int, str(n))); return sum(d) + prod(d)
def ok(n): return any(m + sp(m) == n for m in range(n))
print([k for k in range(111) if ok(k)]) # Michael S. Branicky, Dec 19 2022

a(n) ~ kn with k approximately 1.108374, see comments. - Charles R Greathouse IV, Nov 16 2022

cp's OEIS Frontend

A358512 a(n) is the smallest number k with exactly n divisors that can be written in the form m + digsum(m), for some m (A176995).

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A003052 Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).

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A062028 a(n) = n + sum of the digits of n.

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

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A230099 a(n) = n + (product of digits of n).

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A230094 Numbers that can be expressed as (m + sum of digits of m) in exactly two ways.

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