A062050 -id:A062050 - cp's OEIS frontend

A004759 Binary expansion starts 111.

7, 14, 15, 28, 29, 30, 31, 56, 57, 58, 59, 60, 61, 62, 63, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244

Offset: 1

N. J. A. Sloane

This is the minimal recursive sequence such that a(1)=7, A007814(a(n))= A007814(n) and A010060(a(n))=A010060(n). - Vladimir Shevelev, Apr 23 2009

			30 in binary is 11110, so 30 is in sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..4095

Cf. A004754 (10), A004755 (11), A004756 (100), A004757 (101), A004758 (110).
Cf. A004760, A053644, A062050, A076877.
Cf. A007814, A010060, A103204, A159559, A159560, A159615, A159619, A159629, A159698. -Vladimir Shevelev, Apr 23 2009

import Data.List (transpose)
a004759 n = a004759_list !! (n-1)
a004759_list = 7 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004759_list
-- Reinhard Zumkeller, Dec 03 2015

w = {1, 1, 1}; Select[Range[5, 244], If[# < 2^(Length@ w - 1), True, Take[IntegerDigits[#, 2], Length@ w] == w] &] (* Michael De Vlieger, Aug 10 2016 *)
Sort[FromDigits[#,2]&/@(Flatten[Table[Join[{1,1,1},#]&/@Tuples[{1,0},n],{n,0,5}],1])] (* Harvey P. Dale, Sep 01 2016 *)

PARI
```
a(n)=n+6*2^floor(log(n)/log(2))
```

def A004759(n): return n+(3<Chai Wah Wu, Jul 13 2022

a(2n) = 2a(n), a(2n+1) = 2a(n) + 1 + 6[n==0].
a(n) = n + 6 * 2^floor(log_2(n)) = A004758(n) + A053644(n).
a(n+1) = min{m > a(n): A007814(m) = A007814(n+1) and A010060(m) = A010060(n+1)}. a(2^k) - a(2^k-1) = A103204(k+2), k >= 1. - Vladimir Shevelev, Apr 23 2009
a(2^m+k) = 7*2^m + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016

Edited by Ralf Stephan, Oct 12 2003

A049971 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 2.

Original entry on oeis.org

1, 3, 2, 9, 24, 42, 90, 213, 597, 984, 1974, 3981, 8133, 17037, 37071, 87198, 244557, 401919, 803844, 1607721, 3215613, 6431997, 12866991, 25747038, 51564237, 103443195, 208092192, 421008660, 861332361, 1800360879, 3918287223, 9215926665, 25847419116, 42478911570, 84957823146

Offset: 1

Clark Kimberling

			a(5) = a(1) + a(2) + a(3) + a(4) + a(m) = 1 + 3 + 2 + 9 + a(m) = 15 + a(m). where m = 2*n - 2 - 2^(p+1) and 2^p < n - 1 = 4 <= 2^(p+1). We have p = 1 giving m = 2*5 - 2 - 4 = 4. As a(m) = a(4) = 9, we have a(5) = 15 + 9 = 24. - _David A. Corneth_, Apr 26 2020

David A. Corneth, Table of n, a(n) for n = 1..1000

Cf. A049922 (similar, but with minus a(m/2)), A049923 (similar, but with minus a(m)), A049970 (similar, but with plus a(m/2)).

Cf A062050.

s := proc(n) option remember; `if`(n < 1, 0, a(n) + s(n - 1)) end proc:
a := proc(n) option remember;
`if`(n < 4, [1, 3, 2][n], s(n - 1) + a(-2^ceil(log[2](n - 1)) + 2*n - 2)):
end proc:
seq(a(n), n = 1..40); # Petros Hadjicostas, Apr 25 2020
# Alternative, uses A062050:
a := proc(n) option remember; if n < 4 then [1, 3, 2][n] else
add(a(i), i = 1..n-1 ) + a(2*A062050(n-2)) fi end:
seq(a(n), n = 1..35); # Peter Luschny, Aug 06 2021

a[1] = 1; a[2] = 3; a[3] = 2; a[n_] := a[n] = Sum[a[k], {k, 1, n - 1}] + a[2*n - 2 - 2^Floor[1 + Log[2, n - 2]]]; Table[a[n], {n, 1, 30}] (* Vaclav Kotesovec, Apr 26 2020 *)

lista(nn) = { my(va = vector(nn)); va[1] = 1; va[2] = 3; va[3] = 2; my(sa = vecsum(va)); for (n=4, nn, va[n] = sa + va[2*n - 2 - 2*2^logint(n-2, 2)]; sa += va[n]; ); va; } \\ Michel Marcus, Apr 26 2020 (with nn > 2)

first(n) = {n = max(n, 3); my(res = vector(n), s = 6, p = 1); res[1]  = 1; res[2] = 3; res[3] = 2; for(i = 4, n, if(i - 1 > 1 << (p + 1), p++); res[i] = s + res[2*i-2-2^(p+1)]; s += res[i]) ; res} \\ David A. Corneth, Apr 26 2020

Name edited by Petros Hadjicostas, Apr 25 2020

More terms from David A. Corneth, Apr 26 2020

A092754 a(1)=1, a(2n)=2a(n)+1, a(2n+1)=2a(n)+2.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 21, 22, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 127, 128, 129, 130, 131, 132

Offset: 1

Benoit Cloitre, Apr 13 2004

More generally the sequence b(1)=1, b(2n)=2b(n)+x, b(2n+1)=2b(n)+y is given by the formula b(n)=A053644(n)+x*(n-A053644(n))+y*(A053644(n)-1).

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

Cf. A053644 (x=y=0), A054429(x=-1, y=+1), A062050(x=+1, y=-1).
Cf. A206332 (complement).
Cf. A004754.

a092754 n = if n < 2 then n else 2 * a092754 n' + m + 1
            where (n',m) = divMod n 2
a092754_list = map a092754 [1..]
-- Reinhard Zumkeller, May 07 2012

a(n)=if(n<2,1,if(n%2,a(n-1)+1,a(n/2)*2+1))

PARI
```
a(n) = n + 1<Kevin Ryde, Jun 19 2021
```

a(n) = 2^floor(log(n)/log(2)) + n - 1.

a(n) = A004754(n) - 1. - Rémy Sigrist, May 05 2019

A107681 Repeat(first 2^k numbers with no 2 in ternary representation) for k>0.

Original entry on oeis.org

0, 1, 0, 1, 3, 4, 0, 1, 3, 4, 9, 10, 12, 13, 0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37

Offset: 1

Reinhard Zumkeller, May 20 2005

let A032924(n) = Sum(d(i)*3^i: 0

then .... a(n) = Sum((d(i)-1)*3^i: 0<=i

A032924(n) = A107680(n) + a(n);

A081604 (a(n)) <= A081604(A107680(n)) = A081604(A032924(n)) = A000523(n+1).

			A032924(177) = A107680(177) + a(177),
....... 1420 = ....... 1093 + 327,
.. '1221121' = ... '1111111'+ '110010',
............ = . A003462(7) + A005836(51).

Ruud H.G. van Tol, Table of n, a(n) for n = 1..8190 (12 rows).

Cf. A000523, A005836, A003462, A007089, A032924, A062050, A081604, A107680.

a(n)= fromdigits(binary(n+1-1<Ruud H.G. van Tol, Nov 18 2024

def A107681(n): return int(bin(n+1)[3:],3) # Chai Wah Wu, May 06 2025

a(n) = A005836(A062050(n+1)).

Data corrected by Ruud H.G. van Tol, Nov 18 2024.

A152423 A variation of the Josephus problem, removing every other person, starting with person 1; a(n) is the last person remaining.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 8, 10, 12, 14, 16, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

Offset: 1

Suttapong Wara-asawapati (retsam_krad(AT)hotmail.com), Dec 03 2008

Begin with n people standing in a circle, numbered clockwise 1 through n. Until only one person remains, go around the circle clockwise, removing every other person, starting by removing person 1. a(n) is the number of the last person remaining.

Apparently a(n) = 2*A062050(n-1), n > 1. - Paul Curtz, May 30 2011

			From _Omar E. Pol_, Dec 16 2013: (Start)
It appears that this is also an irregular triangle with row lengths A011782 as shown below:
1;
2;
2,4;
2,4,6,8;
2,4,6,8,10,12,14,16;
2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32;
2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40, 42,44,46,48,50,52,54,56,58,60,62,64;
Right border gives A000079.
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..8192
Eric Weisstein's World of Mathematics, Josephus Problem
Wikipedia, Josephus problem
Index entries for sequences related to the Josephus Problem

The Index to the OEIS lists 21 entries under "Josephus problem". - N. J. A. Sloane, Dec 04 2008

Cf. A000079, A011782, A062050.

a:= n-> 2*n - 2^ceil(log[2](n)):
seq(a(n), n=1..74);  # Alois P. Heinz, Nov 22 2023

A152423[n_]:=2n-2^Ceiling[Log2[n]];Array[A152423,100] (* Paolo Xausa, Nov 23 2023 *)

function F($in){ $a[1] = 1; if($in == 1){ return $a;} $temp =2; for($i=2;$i<=$in;$i++){ $temp+=2; if($temp>$i){ $temp = 2 ; } $answer[] = $temp; } return $answer; } #change $n value for the result $n=5; #sequence store in $answer by using $a = F($n); #to display a(n) echo $a[n];

m=len(bin(n))-3; print(n if 2**m==n else 2*(n-2**m)) # Nicolas Patrois, Apr 19 2021

a(1)=1, a(2)=2; for n > 2, a(n)=2 if n < a(n-1) + 2, otherwise a(n) = a(n-1) + 2.
a(n)=n if n is a power of 2, otherwise a(n)=2*(n-2^m) where m is the exponent of the nearest power of 2 below n. - Nicolas Patrois, Apr 19 2021
a(n) = 2*n - 2^ceiling(log_2(n)). - Alois P. Heinz, Nov 22 2023

Edited by Jon E. Schoenfield, Feb 29 2020

A120241 a(n) = (n - 2^floor(log(n)/log(2)) + 1)-th integer among those positive integers not among the earlier terms of the sequence.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 10, 5, 9, 12, 14, 16, 18, 20, 22, 7, 13, 17, 21, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 11, 19, 25, 29, 33, 37, 41, 45, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 15, 27, 35, 43, 49, 53, 57, 61, 65

Offset: 1

Leroy Quet, Jun 11 2006

This sequence is a permutation of the positive integers. n - 2^floor(log(n)/log(2)) + 1 = A062050(n).

			The first 6 terms of the sequence are 1,2,4,3,6,8. Now 7 - 2^floor(log(7)/log(2)) + 1 = 4. So we want the 4th term of those positive integers not occurring among the first 6 terms of the sequence (i.e., the 4th term among 5,7,9,10,11,...). So a(7) = 10.

Ivan Neretin, Table of n, a(n) for n = 1..8191

Cf. A120242, A062050.

Fold[Append[#1, Complement[Range[Max[#1] + #2], #1][[#2]]] &, {1},
Flatten@Table[Range[2^k], {k, 6}]] (* Ivan Neretin, Sep 24 2021 *)

Extended by Ray Chandler, Jun 19 2006

A353292 a(n) is the number of positive integers k <= n that have at least one common 1-bit with n.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 5, 7, 1, 6, 7, 10, 9, 12, 13, 15, 1, 10, 11, 16, 13, 18, 19, 22, 17, 22, 23, 26, 25, 28, 29, 31, 1, 18, 19, 28, 21, 30, 31, 36, 25, 34, 35, 40, 37, 42, 43, 46, 33, 42, 43, 48, 45, 50, 51, 54, 49, 54, 55, 58, 57, 60, 61, 63, 1, 34, 35, 52, 37

Offset: 0

Rémy Sigrist, Apr 09 2022

See A353293 for the corresponding k's.

			For n = 10:
- we have:
      k   10 AND k
      --  --------
       1         0
       2         2
       3         2
       4         0
       5         0
       6         2
       7         2
       8         8
       9         8
      10        10
- so a(10) = #{2, 3, 6, 7, 8, 9, 10} = 7.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A062050, A080100, A088512, A115378, A353293.

a(n) = { my (h=hammingweight(n), w=#binary(n)); n-2^(w-1)+1 + (2^(h-1)-1)*2^(w-h) }

a(n) = n - A115378(n) for any n > 0.
a(n) = A062050(n) + A088512(n) * A080100(n) for any n > 0.
a(2^k) = 1 for any k >= 0.
a(2^k - 1) = 2^k - 1 for any k >= 0.

A368531 Numbers whose binary indices are all powers of 3, where a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion.

Original entry on oeis.org

0, 1, 4, 5, 256, 257, 260, 261, 67108864, 67108865, 67108868, 67108869, 67109120, 67109121, 67109124, 67109125, 1208925819614629174706176, 1208925819614629174706177, 1208925819614629174706180, 1208925819614629174706181, 1208925819614629174706432

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

For powers of 2 instead of 3 we have A253317.

			The terms together with their binary expansions and binary indices begin:
         0:                           0 ~ {}
         1:                           1 ~ {1}
         4:                         100 ~ {3}
         5:                         101 ~ {1,3}
       256:                   100000000 ~ {9}
       257:                   100000001 ~ {1,9}
       260:                   100000100 ~ {3,9}
       261:                   100000101 ~ {1,3,9}
  67108864: 100000000000000000000000000 ~ {27}
  67108865: 100000000000000000000000001 ~ {1,27}
  67108868: 100000000000000000000000100 ~ {3,27}
  67108869: 100000000000000000000000101 ~ {1,3,27}
  67109120: 100000000000000000100000000 ~ {9,27}
  67109121: 100000000000000000100000001 ~ {1,9,27}
  67109124: 100000000000000000100000100 ~ {3,9,27}
  67109125: 100000000000000000100000101 ~ {1,3,9,27}

Michael De Vlieger, Table of n, a(n) for n = 1..256

A000244 lists powers of 3.
A048793 lists binary indices, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A058891, A062050, A072639, A253317, A326031, A326675, A326702, A367912, A368183, A368109.

Select[Range[0,10000],IntegerQ[Log[3,Times@@Join@@Position[Reverse[IntegerDigits[#,2]],1]]]&]
(* Second program *)
{0}~Join~Array[FromDigits[Reverse@ ReplacePart[ConstantArray[0, Max[#]], Map[# -> 1 &, #]], 2] &[3^(Position[Reverse@ IntegerDigits[#, 2], 1][[;; , 1]] - 1)] &, 255] (* Michael De Vlieger, Dec 29 2023 *)

a(3^n) = 2^(3^n - 1).

A333906 For n >= 2, a(n) = Sum_{k=2..n} prevpower2(k) + nextpower2(k) - 2*k, where prevpower2(k) is the largest power of 2 < k, nextpower2(k) is the smallest power of 2 > k.

Original entry on oeis.org

1, 1, 3, 5, 5, 3, 7, 13, 17, 19, 19, 17, 13, 7, 15, 29, 41, 51, 59, 65, 69, 71, 71, 69, 65, 59, 51, 41, 29, 15, 31, 61, 89, 115, 139, 161, 181, 199, 215, 229, 241, 251, 259, 265, 269, 271, 271, 269, 265, 259, 251, 241, 229, 215, 199, 181, 161

Offset: 2

Ctibor O. Zizka, Apr 09 2020

nonn

Partial sums of b(k) = prevpower2(k) + nextpower2(k) - 2*k; b(k) = 0 for A007283.

			a(2) = (1 + 4 - 2*2) = 1;
a(3) = (1 + 4 - 2*2) + (2 + 4 - 2*3) = 1;
a(4) = (1 + 4 - 2*2) + (2 + 4 - 2*3) + (2 + 8 - 2*4) = 3.

Cf. A000079, A007283, A062050, A080079.

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A004759 Binary expansion starts 111.

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A049971 a(n) = a(1) + a(2) + ... + a(n-1) + a(m) for n >= 4, where m = 2*n - 2 - 2^(p+1) and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1, a(2) = 3, and a(3) = 2.

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A092754 a(1)=1, a(2n)=2a(n)+1, a(2n+1)=2a(n)+2.

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A107681 Repeat(first 2^k numbers with no 2 in ternary representation) for k>0.

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A152423 A variation of the Josephus problem, removing every other person, starting with person 1; a(n) is the last person remaining.

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A120241 a(n) = (n - 2^floor(log(n)/log(2)) + 1)-th integer among those positive integers not among the earlier terms of the sequence.

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