A230093 -id:A230093 - cp's OEIS frontend

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

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

A228085 a(n) = number of distinct k which satisfy n = k + wt(k), where wt(k) (A000120) gives the number of 1's in binary representation of a nonnegative integer k.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 09 2013

wt(k) is also called bitcount(k).
a(n) = number of times n occurs in A092391.
The first 2 occurs at n = A230303(2) = 5 (as we have two solutions A092391(3) = A092391(4) = 5).
The first 3 occurs at n = A230303(3) = 129 (as we have three solutions A092391(123) = A092391(124) = A092391(128) = 129).
The first 4 occurs at n = A230303(4) = 4102, where we have solutions A092391(4091) = A092391(4092) = A092391(4099) = A092391(4100) = 4102.
For n>=1, a(2^n) = a(n-1) since an integer k = m is a solution to n-1 = m + wt(m) if and only if k = 2^n - 1 - m is a solution to 2^n = k + wt(k). - Max Alekseyev, Feb 23 2021

Antti Karttunen, Table of n, a(n) for n = 0..8191
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.
Index entries for Colombian or self numbers and related sequences

A010061 gives the position of zeros, A228082 the positions of nonzeros, A228088 the positions of ones.

Cf. A000120, A010062, A092391, A228086, A228087, A228091 (positions of 2's), A227643, A230058, A230092 (positions of 3's), A230093, A227915 (positions of 4's), A070939, A230303.

a228085 n = length $ filter ((== n) . a092391) [n - a070939 n .. n]
-- Reinhard Zumkeller, Oct 13 2013

For Maple code see A230091. - N. J. A. Sloane, Oct 10 2013
# Find all inverses of m under x -> x + wt(x) - N. J. A. Sloane, Oct 19 2013
A000120 := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120;
F:=proc(m) local ans,lb,n,i;
lb:=m-ceil(log(m+1)/log(2)); ans:=[];
for n from max(1,lb) to m do if (n+wt(n)) = m then ans:=[op(ans),n]; fi; od:
[seq(ans[i],i=1..nops(ans))];
end;

nmax = 8191; Clear[a]; a[_] = 0;
Scan[Set[a[#[[1]]], #[[2]]]&, Tally[Table[n + DigitCount[n, 2, 1], {n, 0, nmax}]]];
a /@ Range[0, nmax] (* Jean-François Alcover, Oct 29 2019 *)
a[n_] := Module[{k, cnt = 0}, For[k = n - Floor[Log[2, n]] - 1, k < n, k++, If[n == k + DigitCount[k, 2, 1], cnt++]]; cnt];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 28 2020 *)

A176995 Numbers that can be written 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

Reinhard Zumkeller, Aug 21 2011

The asymptotic density of this sequence is approximately 0.9022222 (Guaraldo, 1978). - Amiram Eldar, Nov 22 2020

			a(5) = 10 = 5 + (5);
a(87) = 100 = 86 + (8+6);
a(898) = 1000 = 977 + (9+7+7);
a(9017) = 10000 = 9968 + (9+9+6+8).

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).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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.
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.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.

Complement of A003052, range of A062028.

Cf. A062028, A007953, A055642, A230093.

a176995 n = a176995_list !! (n-1)
a176995_list = filter ((> 0) . a230093) [1..]
-- Reinhard Zumkeller, Oct 11 2013, Aug 21 2011

Select[Union[Table[n + Total[IntegerDigits[n]], {n, 77}]], # <= 77 &] (* Jayanta Basu, Jul 27 2013 *)

is_A003052(n)={for(i=1, min(n\2, 9*#digits(n)), sumdigits(n-i)==i && return); n} \\ from A003052
isok(n) = ! is_A003052(n) \\ Michel Marcus, Aug 20 2020

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

A006064 Smallest junction number with n generators.

Original entry on oeis.org

0, 101, 10000000000001, 1000000000000000000000102

Offset: 1

N. J. A. Sloane

Strictly speaking, a junction number is a number n with more than one solution to x+digitsum(x) = n. However, it seems best to start this sequence with n=0, for which there is just one solution, x=0. - N. J. A. Sloane, Oct 31 2013.
a(3) = 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(4) = 10^24 + 102 was conjectured by Narasinga Rao.
a(5) = 10^1111111111124 + 102. - Conjectured by Narasinga Rao, confirmed by Max Alekseyev and N. J. A. Sloane.
a(6) = 10^2222222222224 + 10000000000002. - Max Alekseyev
a(7) = 10^( (10^24 + 10^13 + 115) / 9 ) + 10^13 + 2. - Max Alekseyev
a(8) = 10^( (2*10^24 + 214)/9 ) + 10^24 + 103. - Max Alekseyev

			a(2) = 101 since 101 is the smallest number with two generators: 101 = A062028(91) = A062028(100).
a(4) = 10^24 + 102 = 1000000000000000000000102 has exactly four inverses w.r.t. A062028, namely 999999999999999999999893, 999999999999999999999902, 1000000000000000000000091 and 1000000000000000000000100.

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.
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]
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
Terry Trotter, Charlene numbers
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A230093, A230100, A230303, A230857 (highest power of 10).

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.

a(n) = the smallest m such that there are exactly n solutions to A062028(x)=m.

Edited, a(5)-a(6) added by Max Alekseyev, Jun 01 2011

a(1) added, a(5) corrected, a(7)-a(8) added by Max Alekseyev, Oct 26 2013

A107740 Number of numbers m such that prime(n) = m + (digit sum of m).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 23 2005

a(A049084(A006378(n))) = 0; a(A049084(A048521(n))) > 0. [Corrected by Reinhard Zumkeller, Sep 27 2014]
a(n) <= 2 for n <= 10^5. Conjecture: sequence is bounded.
I would rather conjecture the opposite. Of course a(n) >= m implies n >= A006064(m), having more than A230857(m) digits, i.e., 14, 25 and 1111111111125 digits of n, for a(n) = 3, 4, 5. - M. F. Hasler, Nov 09 2018

			A000040(26) = 101 = 91 + (9 + 1) = 100 + (1 + 0 + 0): a(26) = # {91, 100} = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007953, A000040, A062028, A047791, A107741.

Cf. A006378, A048521, A049084, A055642, A062028.

a107740 n = length [() | let p = a000040 n,
                         m <- [max 0 (p - 9 * a055642 p) .. p - 1],
                         a062028 m == p]
-- Reinhard Zumkeller, Sep 27 2014

Table[p=Prime[n];c=0;i=1;While[iJayanta Basu, May 03 2013 *)

apply( A107740(n)=A230093(prime(n)), [1..150]) \\ M. F. Hasler, Nov 08 2018

a(n) = A230093(prime(n)), i.e.: A107740 = A230093 o A000040. - 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

A230303 Let M(1)=0 and for n >= 2, let B(n)=M(ceiling(n/2))+M(floor(n/2))+2, M(n)=2^B(n)+M(floor(n/2))+1; sequence gives M(n).

Original entry on oeis.org

0, 5, 129, 4102, 87112285931760246646623899502532662132742, 1852673427797059126777135760139006525652319754650249024631321344126610074239106

Offset: 1

N. J. A. Sloane, Oct 24 2013; Mar 26 2014

nonn

M(n) is the smallest value of k such that A228085(k) = n. For example, 129 is the first time a 3 appears in A228085 (and is therefore the first term in A230092). M(4) = 4102 is the first time a 4 appears in A228085 (and is therefore the first term in A227915).

			The terms are a(1) = 0, a(2) = 2^2+0+1, a(3) = 2^7+0+1, a(4) = 2^12+5+1, a(5) = 2^136+5+1, a(6) = 2^160+129+1, a(7) = 2^4233+129+1, a(8) = 2^8206+4102+1, a(9) = 2^k+4102+1 with k=2^136+4110, ... .
The length (in bits) of the n-th term is A230302(n)+1.

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.
Index entries for Colombian or self numbers and related sequences

Cf. A228085, A230092, A227915, A230093, A230302 (for B(n)), A230864.

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.

f:=proc(n) option remember; local B, M;
if n<=1 then RETURN([0,0]);
else
if (n mod 2) = 0 then B:=2*f(n/2)[2]+2;
else B:=f((n+1)/2)[2]+f((n-1)/2)[2]+2; fi;
M:=2^B+f(floor(n/2))[2]+1; RETURN([B,M]); fi;
end proc;
[seq(f(n)[2],n=1..6)];

Define i by 2^(i-1) < n <= 2^i. Then it appears that
a(n) = 2^2^2^...^2^x
a tower of height i+3, containing i+2 2's, where x is in the range 0 < x <= 1.
For example, if n=7, i=3, and
a(7) = 2^4233+130 = 2^2^2^2^2^.88303276...
Note also that i+2 = A230864(a(n)).

a(1)-a(8) were found by Donovan Johnson, Oct 22 2013.

A230639 Let M(1)=0 and for n>1, B(n)=(M(ceiling(n/2))+M(floor(n/2))+2)/2, M(n)=3^B(n)+M(floor(n/2))+1. This sequence gives B(n).

Original entry on oeis.org

1, 3, 5, 17, 29, 139, 249, 64570209, 129140169, 34315253252541, 68630377364913, 1044297913696328396542704032390321722034449074468444246791788357605, 2088595827392656793085408064780643444068898148936888424953199350297

Offset: 2

N. J. A. Sloane, Oct 31 2013

The largest power of 3 in M(n) = A230640(n).

Max Alekseyev, Table n, expression for a(n) for n=2..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.
Index entries for Colombian or self numbers and related sequences

Cf. A230093, A230640 (for M(n)).

Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1)

f:=proc(n) option remember; local B, M;
if n<=1 then RETURN([0, 0]);
else
B:=(f(ceil(n/2))[2] + f(floor(n/2))[2] + 2)/2;
M:=3^B+f(floor(n/2))[2]+1; RETURN([B, M]); fi;
end proc;
[seq(f(n)[1], n=1..9)];

Terms a(10) onward from Max Alekseyev, Nov 02 2013

A227915 Numbers of the form k + wt(k) for exactly four distinct k, where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

4102, 12295, 20487, 28680, 36871, 45064, 53256, 61449, 69639, 77832, 86024, 94217, 102408, 110601, 118793, 126986, 135175, 143368, 151560, 159753, 167944, 176137, 184329, 192522, 200712, 208905, 217097, 225290, 233481, 241674, 249866, 258059, 266247, 274440, 282632, 290825, 299016, 307209, 315401, 323594, 331784, 339977

Offset: 1

Reinhard Zumkeller, Oct 13 2013

Numbers occurring exactly four times in A092391: A228085(a(n)) = 4. For the first number that appears k times, see A230303.

			a(1) = 4102, the four k with A092391(k) = 4102 being:
4091 = '111111111011', A000120(4091) = 11, 4091 + 11 = 4102;
4092 = '111111111100', A000120(4092) = 12, 4092 + 10 = 4102;
4099 = '1000000000011', A000120(4099) = 3, 4099 + 3 = 4102;
4100 = '1000000000100', A000120(4100) = 2, 4100 + 2 = 4102.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Cf. A092391, A228085, A230092, A230091, A228088, A010061, A230093, A230303.

a227915 n = a227915_list !! (n-1)
a227915_list = filter ((== 4) . a228085) [1..]

cp's OEIS Frontend

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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A228085 a(n) = number of distinct k which satisfy n = k + wt(k), where wt(k) (A000120) gives the number of 1's in binary representation of a nonnegative integer k.

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A176995 Numbers that can be written as (m + sum of digits of m) for some m.

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A006064 Smallest junction number with n generators.

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A107740 Number of numbers m such that prime(n) = m + (digit sum of m).

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