A228088 -id:A228088 - cp's OEIS frontend

A010061 Binary self or Colombian numbers: numbers that cannot be expressed as the sum of distinct terms of the form 2^k+1 (k>=0), or equivalently, numbers not of form m + sum of binary digits of m.

1, 4, 6, 13, 15, 18, 21, 23, 30, 32, 37, 39, 46, 48, 51, 54, 56, 63, 71, 78, 80, 83, 86, 88, 95, 97, 102, 104, 111, 113, 116, 119, 121, 128, 130, 133, 135, 142, 144, 147, 150, 152, 159, 161, 166, 168, 175, 177, 180, 183, 185, 192, 200, 207, 209, 212, 215, 217

Offset: 1

Leonid Broukhis

No two consecutive values appear in this sequence (see Links). - Griffin N. Macris, May 31 2020

The asymptotic density of this sequence is (1/8) * (2 - Sum_{n>=1} 1/2^a(n))^2 = 0.252660... (A242403). - Amiram Eldar, Nov 28 2020

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24, pp. 179-180.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 384-386.
G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers, Bolletino U. M. I. (7) 9-A (1995), 143-148.

Antti Karttunen, 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 [math.NT], 2021-2022; 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.
Griffin N. Macris, Proof that no consecutive self numbers exist, 2020.
G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers. II, Bollettino dell'Unione Matematica Italiana, 2-A (1999), 397-399.
Index entries for Colombian or self numbers and related sequences

Complement of A228082, or equally, numbers which do not occur in A092391. Gives the positions of zeros (those occurring after a(0)) in A228085-A228087 and positions of ones in A227643. Leftmost column of A228083. Base-10 analog: A003052.
Cf. A010062, A055938, A230091, A230092, A230058, A242403.
Cf. A228088, A227915, A232228.

a010061 n = a010061_list !! (n-1)
a010061_list = filter ((== 0) . a228085) [1..]
-- Reinhard Zumkeller, Oct 13 2013

# For Maple code see A230091. - N. J. A. Sloane, Oct 10 2013

Table[n + Total[IntegerDigits[n, 2]], {n, 0, 300}] // Complement[Range[Last[#]], #]& (* Jean-François Alcover, Sep 03 2013 *)

/* Gen(n, b) returns a list of the generators of n in base b. Written by Max Alekseyev (see Alekseyev et al., 2021).
For example, Gen(101, 10) returns [91, 101]. - N. J. A. Sloane, Jan 02 2022 */
{ Gen(u, b=10) = my(d, m, k);
  if(u<0 || u==1, return([]); );
  if(u==0, return([0]); );
  d = #digits(u, b)-1;
  m = u\b^d;
  while( sumdigits(m, b) > u - m*b^d,
    m--;
    if(m==0, m=b-1; d--; );
  );
  k = u - m*b^d - sumdigits(m, b);
  vecsort( concat( apply(x->x+m*b^d, Gen(k, b)),
                   apply(x->m*b^d-1-x, Gen((b-1)*d-k-2, b)) ) );
}

More terms from Antti Karttunen, Aug 17 2013

Better definition from Matthew C. Russell, Oct 08 2013

A092391 a(n) = n + wt(n), where wt(n) = A000120(n) = binary weight of n.

Original entry on oeis.org

0, 2, 3, 5, 5, 7, 8, 10, 9, 11, 12, 14, 14, 16, 17, 19, 17, 19, 20, 22, 22, 24, 25, 27, 26, 28, 29, 31, 31, 33, 34, 36, 33, 35, 36, 38, 38, 40, 41, 43, 42, 44, 45, 47, 47, 49, 50, 52, 50, 52, 53, 55, 55, 57, 58, 60, 59, 61, 62, 64, 64, 66, 67, 69, 65, 67, 68, 70, 70, 72, 73, 75

Offset: 0

Reinhard Zumkeller, May 08 2004

Reinhard Zumkeller and Donovan Johnson, Table of n, a(n) for n = 0..10000 (terms up to a(1023) from Reinhard Zumkeller)
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 numbers not occurring in this sequence. A228082 gives the terms of this sequence sorted into ascending order, with duplicates removed. A228085(n) gives the number of times n occurs in this sequence.
Cf. A000120, A228083, A228086, A228087, A228091, A011371, A230058, A230092, A062028.
Cf. A228088, A230091, A227915, A230300.

a092391 n = n + a000120 n  -- Reinhard Zumkeller, May 13 2012

Table[n + Total[IntegerDigits[n, 2]], {n, 0, 100}] (* Jean-François Alcover, Sep 03 2013 *)

A092391(n)=n+hammingweight(n) \\ M. F. Hasler, Oct 05 2013

def a(n): return n + bin(n).count("1")
print([a(n) for n in range(72)]) # Michael S. Branicky, May 26 2022

a(n) = n + A000120(n).
A010062(n+1) = a(A010062(n)).
G.f.: (1/(1 - x))*Sum_{k>=0} (2^k + 1)*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jul 23 2017

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 *)

A228082 Numbers that are of the form k + sum of binary digits of k for some nonnegative integer k.

Original entry on oeis.org

0, 2, 3, 5, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 20, 22, 24, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 38, 40, 41, 42, 43, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 81, 82, 84, 85, 87, 89, 90

Offset: 1

Antti Karttunen, Aug 09 2013

Complement of A010061.
Obtained when A092391 is sorted and duplicates are removed.
The asymptotic density of this sequence is 1 - (1/8) * (Sum_{n>=1} 1/2^a(n))^2 = 1 - A242403 = 0.747339... - Amiram Eldar, Nov 28 2020

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24, pp. 179-180.
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, Bolletino U. M. I. (7) 9-A (1995), 143-148.

Antti Karttunen, Table of n, a(n) for n = 1..8192
G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers. II, Bollettino dell'Unione Matematica Italiana 2-A (1999), 397-399.
Index entries for Colombian or self numbers and related sequences

Numbers that occur to the right of the leftmost column of A228083. Positions of nonzeros in A228085. A superset of A228088.
Cf. also A227643, A005187, A230091, A230092, A227915, A242403.
The even terms are the first row of A350601.

a228082 n = a228082_list !! (n-1)
a228082_list = 0 : filter ((> 0) . a228085) [1..]
-- Reinhard Zumkeller, Oct 13 2013

Table[n + Total[IntegerDigits[n, 2]], {n, 0, 100}] // Union (* Jean-François Alcover, Sep 03 2013 *)

A230641 a(n) = n + (sum of digits in base-3 representation of n).

Original entry on oeis.org

0, 2, 4, 4, 6, 8, 8, 10, 12, 10, 12, 14, 14, 16, 18, 18, 20, 22, 20, 22, 24, 24, 26, 28, 28, 30, 32, 28, 30, 32, 32, 34, 36, 36, 38, 40, 38, 40, 42, 42, 44, 46, 46, 48, 50, 48, 50, 52, 52, 54, 56, 56, 58, 60, 56, 58, 60, 60, 62, 64, 64, 66, 68, 66, 68, 70, 70, 72, 74, 74, 76, 78, 76, 78, 80, 80, 82, 84, 84, 86, 88, 82

Offset: 0

N. J. A. Sloane, Oct 31 2013

The image of this sequence is the set of nonnegative even numbers (A005843). Joshi (1973) proved that the sequence of base-q self numbers (analogous to A003052) is the sequence of odd numbers (A005408) for all odd q. - Amiram Eldar, Nov 28 2020

V. S. Joshi, Ph.D. dissertation, Gujarat Univ., Ahmedabad (India), October, 1973.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

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

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

Cf. A228088, A010061, A230091, A230092, A227915.

a230641 n = a053735 n + n  -- Reinhard Zumkeller, May 19 2015

Table[n + Plus @@ IntegerDigits[n, 3], {n, 0, 100}] (* Amiram Eldar, Nov 28 2020 *)

a(n) = n + A053735(n). - Amiram Eldar, Nov 28 2020

A230092 Numbers of the form k + wt(k) for exactly three distinct k, where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

129, 134, 386, 391, 515, 518, 642, 647, 899, 904, 1028, 1030, 1154, 1159, 1411, 1416, 1540, 1543, 1667, 1672, 1924, 1929, 2178, 2183, 2435, 2440, 2564, 2567, 2691, 2696, 2948, 2953, 3077, 3079, 3203, 3208, 3460, 3465, 3589, 3592, 3716, 3721, 3973, 3978, 4226

Offset: 1

N. J. A. Sloane, Oct 10 2013

The positions of entries equal to 3 in A228085, or numbers that appear exactly thrice in A092391.

Numbers that can be expressed as the sum of distinct terms of the form 2^n+1, n=0,1,... in exactly three ways.

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

Cf. A000120, A010061, A228088, A230058, A230091.

Cf. A227915.

a230092 n = a230092_list !! (n-1)
a230092_list = filter ((== 3) . a228085) [1..]
-- Reinhard Zumkeller, Oct 13 2013

Maple
```
For Maple code see A230091.
```

nt = 1000; (* number of terms to produce *)
S[kmax_] := S[kmax] = Table[k + Total[IntegerDigits[k, 2]], {k, 0, kmax}] // Tally // Select[#, #[[2]] == 3&][[All, 1]]& // PadRight[#, nt]&;
S[nt];
S[kmax = 2 nt];
While[S[kmax] =!= S[kmax/2], kmax *= 2];
S[kmax] (* Jean-François Alcover, Mar 04 2023 *)

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..]

A228086 a(n) is the least k which satisfies n = k + bitcount(k), or 0 if no such k exists. Here bitcount(k) (or wt(k), A000120) gives the number of 1's in binary representation of nonnegative integer k.

Original entry on oeis.org

0, 0, 1, 2, 0, 3, 0, 5, 6, 8, 7, 9, 10, 0, 11, 0, 13, 14, 0, 15, 18, 0, 19, 0, 21, 22, 24, 23, 25, 26, 0, 27, 0, 29, 30, 33, 31, 0, 35, 0, 37, 38, 40, 39, 41, 42, 0, 43, 0, 45, 46, 0, 47, 50, 0, 51, 0, 53, 54, 56, 55, 57, 58, 0, 59, 64, 61, 62, 66, 63, 67, 0

Offset: 0

Antti Karttunen, Aug 09 2013

nonn

A083058(n)+1 gives a lower bound for nonzero terms, n-1 an upper bound.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for Colombian or self numbers and related sequences

Cf. A228087, A228085, A335599. A010061 gives the positions of zeros after a(0). The union of A010061 and A228088 gives the positions where a(n) = A228087(n).

Cf. also A213723, A227643.

a[n_] := Module[{k}, For[k = n - Floor[Log[2, n]] - 1, k < n, k++, If[n == k + DigitCount[k, 2, 1], Return[k]]]; 0];
a /@ Range[0, 1000]; (* Jean-François Alcover, Nov 28 2020 *)

(define (A228086 n) (if (zero? n) n (let loop ((k (+ (A083058 n) 1))) (cond ((> k n) 0) ((= n (A092391 k)) k) (else (loop (+ 1 k)))))))

A230091 Numbers of the form k + wt(k) for exactly two distinct k, where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

5, 14, 17, 19, 22, 31, 33, 36, 38, 47, 50, 52, 55, 64, 67, 70, 79, 82, 84, 87, 96, 98, 101, 103, 112, 115, 117, 120, 131, 132, 143, 146, 148, 151, 160, 162, 165, 167, 176, 179, 181, 184, 193, 196, 199, 208, 211, 213, 216, 225, 227, 230, 232, 241, 244, 246, 249, 258, 260, 262, 271, 274, 276, 279, 288, 290, 293, 295

Offset: 1

N. J. A. Sloane, Oct 10 2013

The positions of entries equal to 2 in A228085, or numbers that appear exactly twice in A092391.

Numbers that can be expressed as the sum of distinct terms of the form 2^n+1, n=0,1,... in exactly two ways.

			5 = 3 + 2 = 4 + 1, so 5 is in this list.

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

Cf. A000120, A010061, A092391, A228088, A228085, A230058, A230092.

Cf. A227915.

a230091 n = a230091_list !! (n-1)
a230091_list = filter ((== 2) . a228085) [1..]
-- Reinhard Zumkeller, Oct 13 2013

# Maple code for A000120, A092391, A228085, A010061, A228088, A230091, A230092
with(LinearAlgebra):
read transforms;
wt := 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: # A000120
M:=1000;
lis1:=Array(0..M);
lis2:=Array(0..M);
ctmax:=4;
for i from 0 to ctmax do ct[i]:=Array(0..M); od:
for n from 0 to M do
m:=n+wt(n);
lis1[n]:=m;
if (m <= M) then lis2[m]:=lis2[m]+1; fi;
od:
t1:=[seq(lis1[i],i=0..M)]; # A092391
t2:=[seq(lis2[i],i=0..M)]; # A228085
COMPl(t1); # A010061
for i from 1 to M do h:=lis2[i];
if h <= ctmax then ct[h]:=[op(ct[h]),i]; fi; od:
len:=nops(ct[0]); [seq(ct[0][i],i=1..len)]; # A010061 again
len:=nops(ct[1]); [seq(ct[1][i],i=1..len)]; # A228088
len:=nops(ct[2]); [seq(ct[2][i],i=1..len)]; # A230091
len:=nops(ct[3]); [seq(ct[3][i],i=1..len)]; # A230092

nt = 100; (* number of terms to produce *)
S[kmax_] := S[kmax] = Table[k + Total[IntegerDigits[k, 2]], {k, 0, kmax}] // Tally // Select[#, #[[2]] == 2&][[All, 1]]& // PadRight[#, nt]&;
S[nt];
S[kmax = 2 nt];
While[S[kmax] =!= S[kmax/2], kmax *= 2];
S[kmax] (* Jean-François Alcover, Mar 04 2023 *)

A228087 a(n) = largest k which satisfies n = k + bitcount(k), or 0 if no such k exists. Here bitcount(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 0, 5, 6, 8, 7, 9, 10, 0, 12, 0, 13, 16, 0, 17, 18, 0, 20, 0, 21, 22, 24, 23, 25, 26, 0, 28, 0, 32, 30, 33, 34, 0, 36, 0, 37, 38, 40, 39, 41, 42, 0, 44, 0, 45, 48, 0, 49, 50, 0, 52, 0, 53, 54, 56, 55, 57, 58, 0, 60, 64, 61, 65, 66, 63, 68, 0

Offset: 0

Antti Karttunen, Aug 09 2013

nonn

A083058(n)+1 gives a lower bound for nonzero terms, n-1 an upper bound.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for Colombian or self numbers and related sequences

Cf. A228086, A228085. A010061 gives the positions of zeros after a(0). The union of A010061 and A228088 gives the positions where a(n) = A228086(n).

Cf. also A213724, A227643.

(define (A228087 n) (let loop ((k n)) (cond ((<= k (A083058 n)) 0) ((= n (A092391 k)) k) (else (loop (- k 1))))))

cp's OEIS Frontend

A010061 Binary self or Colombian numbers: numbers that cannot be expressed as the sum of distinct terms of the form 2^k+1 (k>=0), or equivalently, numbers not of form m + sum of binary digits of m.

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A092391 a(n) = n + wt(n), where wt(n) = A000120(n) = binary weight 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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A228082 Numbers that are of the form k + sum of binary digits of k for some nonnegative integer k.

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A230641 a(n) = n + (sum of digits in base-3 representation of n).

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A230092 Numbers of the form k + wt(k) for exactly three distinct k, where wt(k) = A000120(k) is the binary weight of k.

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

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