A228091 -id:A228091 - cp's OEIS frontend

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

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 *)

A228083 Table of binary Self-numbers and their descendants; square array T(r,c), with row r>=1, column c>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 5, 7, 8, 13, 7, 10, 9, 16, 15, 10, 12, 11, 17, 19, 18, 12, 14, 14, 19, 22, 20, 21, 14, 17, 17, 22, 25, 22, 24, 23, 17, 19, 19, 25, 28, 25, 26, 27, 30, 19, 22, 22, 28, 31, 28, 29, 31, 34, 32, 22, 25, 25, 31, 36, 31, 33, 36, 36, 33, 37

Offset: 1

Antti Karttunen, Aug 09 2013

			The top-left corner of the square array:
   1,  2,  3,  5,  7, 10, 12, 14, ...
   4,  5,  7, 10, 12, 14, 17, 19, ...
   6,  8,  9, 11, 14, 17, 19, 22, ...
  13, 16, 17, 19, 22, 25, 28, 31, ...
  15, 19, 22, 25, 28, 31, 36, 38, ...
  18, 20, 22, 25, 28, 31, 36, 38, ...
  21, 24, 26, 29, 33, 35, 38, 41, ...
  23, 27, 31, 36, 38, 41, 44, 47, ...
  ...
The non-initial terms on each row are obtained by adding to the preceding term the number of 1-bits in its binary representation (A000120).

First column: A010061. First row: A010062. Transpose: A228084. See A151942 for decimal analog.

Cf. also A000120, A092391, A227643, A228082, A228085, A228086, A228087, A228088, A228091, A218254.

nmax0 = 100;
nmax := Length[col[1]];
col[1] = Table[n + DigitCount[n, 2, 1], {n, 0, nmax0}] // Complement[Range[Last[#]], #]&;
col[k_] := col[k] = col[k - 1] + DigitCount[col[k-1], 2, 1];
T[n_, k_] := col[k][[n]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 28 2020 *)

T(r,1) are those numbers not of form n + sum of binary digits of n (binary Self numbers) = A010061(r);

T(r,c) = T(r,c-1) + sum of binary digits of T(r,c-1) = A092391(T(r,c-1)).

A228236 Numbers k for which a sum k+bitcount(k) can be also obtained as a sum k2 +bitcount(k2) for some other k2<>k . Here bitcount(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.

Original entry on oeis.org

3, 4, 11, 12, 14, 15, 16, 17, 19, 20, 27, 28, 29, 31, 32, 34, 35, 36, 43, 44, 46, 47, 48, 49, 51, 52, 59, 60, 62, 65, 67, 68, 75, 76, 78, 79, 80, 81, 83, 84, 91, 92, 93, 95, 96, 98, 99, 100, 107, 108, 110, 111, 112, 113, 115, 116, 123, 124, 125, 126, 127, 128

Offset: 1

Antti Karttunen, Aug 17 2013

nonn

In other words, numbers k such that A228085(A092391(k)) > 1.

			0 is not in this sequence because the sum 0+A000120(0)=0 cannot be obtained with any other value of k than k=0.
1 is not in this sequence because the sum 1+A000120(1)=2 cannot be obtained with any other value of k than k=1.
2 is not in this sequence because the sum 2+A000120(2)=3 cannot be obtained with any other value of k than k=2.
3 IS in this sequence because the sum 3+A000120(3)=5 can also be obtained with value k=4, as also 4+A000120(4)=5, and thus also 4 is in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A228090. Subsets: A228091, A228237. Cf. also A092391, A228085.

A228237 Numbers n for which there exists such a natural number k > n that k + bitcount(k) = n + bitcount(n), where bitcount(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.

Original entry on oeis.org

3, 11, 14, 15, 19, 27, 29, 31, 35, 43, 46, 47, 51, 59, 62, 67, 75, 78, 79, 83, 91, 93, 95, 99, 107, 110, 111, 115, 123, 124, 125, 126, 127, 131, 139, 142, 143, 147, 155, 157, 159, 163, 171, 174, 175, 179, 187, 190, 195, 203, 206, 207, 211, 219, 221, 223, 227

Offset: 1

Antti Karttunen, Sep 11 2013

nonn

In other words, all such terms A228236(n) which satisfy A228236(n) < A228087(A092391(A228236(n))).

Note: 124 is the first term that occurs both here and in A228091.

			For cases 0 + A000120(0) = 0, 1 + A000120(1) = 2, 2 + A000120(2) = 3 there are no larger solutions yielding the same result.
However, for 3 + A000120(3) = 5 there is a larger solution yielding the same result, namely 4 + A000120(4) = 5, thus 3 is the first term of this sequence.
Next time this occurs for 11, as 11 + A000120(11) = 14 = 12 + A000120(12), and 12 > 11.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subset of A228236. Cf. also A228091.

cp's OEIS Frontend

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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A228083 Table of binary Self-numbers and their descendants; square array T(r,c), with row r>=1, column c>=1, read by antidiagonals.

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A228236 Numbers k for which a sum k+bitcount(k) can be also obtained as a sum k2 +bitcount(k2) for some other k2<>k . Here bitcount(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.

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A228237 Numbers n for which there exists such a natural number k > n that k + bitcount(k) = n + bitcount(n), where bitcount(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.

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