cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Antti Karttunen, Aug 09 2013

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • Scheme
    (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

Views

Author

N. J. A. Sloane, Oct 10 2013

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A000120, A010061, A092391, A228088, A228085, A230058, A230092.
Cf. A227915.

Programs

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

Views

Author

Antti Karttunen, Aug 09 2013

Keywords

Comments

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

Links

Crossrefs

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.

Programs

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

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

Views

Author

Antti Karttunen, Aug 09 2013

Keywords

Examples

			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).

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

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)).

A230058 Numbers of the form k + wt(k) for at least 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, 129, 131, 132, 134, 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
Offset: 1

Views

Author

Matthew C. Russell, Oct 07 2013

Keywords

Comments

The positions of entries greater than 1 in A228085, or numbers that appear multiple times in A092391.
Numbers that can be expressed as the sum of distinct terms of the form 2^n+1, n=0,1,... in multiple ways.

Examples

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

Links

Crossrefs

Cf. A000120, A010061, A228088, A230091, A230092.

Programs

  • Mathematica
    Sort[Transpose[Select[Tally[Table[k + Total[IntegerDigits[k, 2]], {k, 0, 300}]], #[[2]] > 1 &]][[1]]] (* T. D. Noe, Oct 09 2013 *)

A228091 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

4, 12, 16, 17, 20, 28, 32, 34, 36, 44, 48, 49, 52, 60, 65, 68, 76, 80, 81, 84, 92, 96, 98, 100, 108, 112, 113, 116, 124, 128, 129, 130, 131, 132, 140, 144, 145, 148, 156, 160, 162, 164, 172, 176, 177, 180, 188, 193, 196, 204, 208, 209, 212, 220, 224, 226, 228
Offset: 1

Views

Author

Antti Karttunen, Aug 09 2013

Keywords

Comments

In other words, all such terms A228236(n) which satisfy A228236(n) > A228086(A092391(A228236(n))), which means that the sequence contains all natural numbers n such that A228085(A092391(n)) > 1 and n > A228086(A092391(n)).
Note: 124 is the first term that occurs both here and in A228237.

Examples

			For cases 0 + A000120(0) = 0, 1 + A000120(1) = 2, 2 + A000120(2) = 3, 3 + A000120(3) = 5 there are no smaller solutions yielding the same result.
However, for 4 + A000120(4) = 5, we already saw the case 3+A000120(3) giving the same result, thus 4 is the first term of this sequence.
Next time this occurs for 12, as 12 + A000120(12) = 14 = 11 + A000120(11), and 11 < 12.

Links

Crossrefs

Subset of A228236. Cf. also A228237. Complement of this sequence gives the nonzero terms of A228086 in ascending order.
Cf. A228085, A228086, A092391.

A228090 Numbers k for which a sum k + bitcount(k) cannot be obtained as a sum k2 + bitcount(k2) for any 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

0, 1, 2, 5, 6, 7, 8, 9, 10, 13, 18, 21, 22, 23, 24, 25, 26, 30, 33, 37, 38, 39, 40, 41, 42, 45, 50, 53, 54, 55, 56, 57, 58, 61, 63, 64, 66, 69, 70, 71, 72, 73, 74, 77, 82, 85, 86, 87, 88, 89, 90, 94, 97, 101, 102, 103, 104, 105, 106, 109, 114, 117, 118, 119, 120
Offset: 1

Views

Author

Antti Karttunen, Aug 17 2013

Keywords

Comments

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

Examples

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

Links

Crossrefs

Sequence A228089 sorted into ascending order. Complement: A228236.
Cf. also A092391, A228085, A228088.

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

Views

Author

Antti Karttunen, Aug 17 2013

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

A230300 a(n) = n + wt(n-1), where wt() = A000120() is the binary weight.

Original entry on oeis.org

1, 3, 4, 6, 6, 8, 9, 11, 10, 12, 13, 15, 15, 17, 18, 20, 18, 20, 21, 23, 23, 25, 26, 28, 27, 29, 30, 32, 32, 34, 35, 37, 34, 36, 37, 39, 39, 41, 42, 44, 43, 45, 46, 48, 48, 50, 51, 53, 51, 53, 54, 56, 56, 58, 59, 61, 60, 62, 63, 65, 65, 67, 68, 70, 66, 68, 69, 71, 71, 73, 74, 76, 75, 77, 78, 80, 80, 82, 83, 85, 83, 85
Offset: 1

Views

Author

N. J. A. Sloane, Oct 23 2013

Keywords

Comments

A228085(m-1) gives the number of times m occurs in this sequence.

Links

Crossrefs

Cf. A000120, A092391, A228085, A230301 (complement).

Programs

  • Maple
    a:= n-> n+add(i,i=Bits[Split](n-1)):
seq(a(n), n=1..82);  # Alois P. Heinz, Jul 05 2024
  • Mathematica
    Table[n+Total[IntegerDigits[n-1,2]],{n,100}] (* Harvey P. Dale, May 20 2015 *)
  • PARI
    a(n) = n + hammingweight(n-1); \\ Michel Marcus, Jul 05 2024

A230302 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 B(n).

Original entry on oeis.org

2, 7, 12, 136, 260, 4233, 8206, 87112285931760246646623899502532662136846, 174224571863520493293247799005065324265486, 1852673427797059126777135760139006525739432040582009271277945243629142736371850, 3705346855594118253554271520278013051304639509300498049262642688253220148478214
Offset: 2

Views

Author

N. J. A. Sloane, Oct 24 2013

Keywords

Comments

a(n) is the leading power of 2 in M(n) = A230303(n).

Examples

			The terms after 8206 are 2^136+4110, 2^137+14, 2^260+2^136+136, 2^261+262, 2^4233+2^260+260, ... (see also A230303).

Links

Crossrefs

Cf. A228085, A230093, A230303 (for M(n)).

Programs

  • Maple
    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)[1],n=1..7)];

Extensions

a(11) corrected, expressions for a(2)-a(100) added by Max Alekseyev, Nov 02 2013
Previous Showing 11-20 of 22 results. Next

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