A087117 -id:A087117 - cp's OEIS frontend

A330036 The length of the largest run of 0's in the binary expansion of n + the length of the largest run of 1's in the binary expansion of n.

1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 2, 3, 4, 3, 4, 4, 5, 4, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 5, 4, 5, 5, 6, 5, 4, 5, 3, 3, 4, 5, 4, 3, 2, 3, 4, 3, 4, 5, 6, 5, 4, 4, 4, 3, 3, 4, 6, 5, 4, 4, 6, 5, 6, 6, 7, 6, 5, 6, 4, 4, 5, 6, 4, 3, 3, 4, 4, 4, 5, 6, 5, 4, 3

Offset: 0

Joshua Oliver, Nov 27 2019

All numbers appear in this sequence. The number of 1's in the n-th Mersenne number (A000225) is n and the number of 0's in the n-th Mersenne number is 0. 0+n=n. See formula.

			   n [binary n ]  A087117(n) + A038374(n) = a(n)
   0 [       0 ]  1          + 0          = 1
   1 [       1 ]  0          + 1          = 1
   2 [     1 0 ]  1          + 1          = 2
   3 [     1 1 ]  0          + 2          = 2
   4 [   1 0 0 ]  2          + 1          = 3
   5 [   1 0 1 ]  1          + 1          = 2
   6 [   1 1 0 ]  1          + 2          = 3
   7 [   1 1 1 ]  0          + 3          = 3
   8 [ 1 0 0 0 ]  3          + 1          = 4
   9 [ 1 0 0 1 ]  2          + 1          = 3
  10 [ 1 0 1 0 ]  1          + 1          = 2
  11 [ 1 0 1 1 ]  1          + 2          = 3
  12 [ 1 1 0 0 ]  2          + 2          = 4
  13 [ 1 1 0 1 ]  1          + 2          = 3
  14 [ 1 1 1 0 ]  1          + 3          = 4
  15 [ 1 1 1 1 ]  0          + 4          = 4

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000120, A000225, A007088, A023416, A038374, A087117.

f:= proc(n) local L;
  L:= convert(n,base,2);
  max(map(nops,[ListTools:-Split(`=`,L,1)]))+max(map(nops,[ListTools:-Split(`=`,L,0)]))
end proc:
map(f, [$0..100]); # Robert Israel, Apr 06 2020

Table[Sum[Max[Differences[Position[Flatten@{k,IntegerDigits[n,2],k},k]]],{k,0,1}]-2,{n,0,82}]

a(n) = A087117(n) + A038374(n).

a(A000225(n)) = n for n > 0.

A090047 Length of longest contiguous block of 0's in binary expansion of n^2.

Original entry on oeis.org

1, 0, 2, 2, 4, 2, 2, 3, 6, 3, 2, 2, 4, 2, 3, 4, 8, 4, 3, 2, 4, 2, 2, 4, 6, 3, 2, 2, 4, 2, 4, 5, 10, 5, 4, 2, 4, 2, 2, 3, 6, 3, 2, 2, 4, 2, 4, 4, 8, 4, 3, 3, 4, 2, 2, 3, 6, 3, 2, 2, 4, 3, 5, 6, 12, 6, 5, 3, 4, 2, 2, 3, 6, 3, 2, 2, 4, 2, 3, 4, 8, 4, 3, 2, 4, 4, 2, 3, 6, 3, 2, 6, 4, 4, 4, 5, 10, 5, 4, 2, 4, 2

Offset: 0

Reinhard Zumkeller, Nov 20 2003

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000290, A023416, A087117, A090001, A090046, A090048, A090049.

a[n_] := If[n <= 1, 1-n, Max[Length /@ Split[IntegerDigits[n^2, 2]][[2 ;; -1 ;; 2]]]]; Array[a, 100, 0] (* Amiram Eldar, Jul 28 2025 *)

a(n) = A087117(A000290(n)).

A090048 Length of longest contiguous block of 0's in binary expansion of n-th triangular number.

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 2, 2, 1, 1, 4, 2, 1, 2, 3, 3, 2, 1, 1, 2, 2, 1, 3, 2, 3, 1, 1, 2, 2, 3, 4, 4, 3, 2, 2, 2, 1, 2, 4, 2, 1, 4, 2, 1, 6, 4, 3, 3, 2, 2, 2, 3, 2, 2, 6, 3, 2, 1, 1, 2, 3, 4, 5, 5, 4, 3, 3, 2, 2, 2, 2, 3, 3, 1, 3, 1, 1, 6, 3, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 0, 5, 4, 3, 3, 4, 4, 3, 2, 2, 2, 5

Offset: 0

Reinhard Zumkeller, Nov 20 2003

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000217, A023416, A087117, A090002, A090047, A090049.

a[n_] := Module[{d = IntegerDigits[n*(n + 1)/2, 2]}, If[! MemberQ[d, 0], 0, Max[Length /@ Split[d][[2 ;; -1 ;; 2]]]]]; a[0] = 1; Array[a, 102, 0] (* Amiram Eldar, Jul 29 2025 *)

a(n) = A087117(A000217(n)).

A090049 Length of longest contiguous block of 0's in binary expansion of n^3.

Original entry on oeis.org

1, 0, 3, 1, 6, 1, 3, 1, 9, 2, 3, 2, 6, 3, 3, 2, 12, 3, 3, 2, 6, 4, 3, 4, 9, 4, 3, 3, 6, 3, 3, 3, 15, 4, 3, 2, 6, 3, 3, 2, 9, 4, 4, 2, 6, 3, 4, 3, 12, 3, 4, 6, 6, 3, 3, 3, 9, 2, 3, 4, 6, 2, 3, 4, 18, 5, 4, 2, 6, 6, 3, 4, 9, 2, 3, 2, 6, 1, 3, 4, 12, 6, 4, 3, 6, 2, 3, 5, 9, 5, 3, 3, 6, 6, 3, 3, 15, 4, 3

Offset: 0

Reinhard Zumkeller, Nov 20 2003

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000578, A007814, A023416, A070939, A087117, A090003, A090047, A090048.

f:= proc(n) local R;
  R:= convert(2*n^3+1,base,2);
  R:= select(t -> R[t]=1, [$1..nops(R)]);
  max(R[2..-1]-R[1..-2])-1
end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Mar 10 2020

a[n_] := Module[{d = IntegerDigits[n^3, 2]}, If[! MemberQ[d, 0], 0, Max[Length /@ Split[d][[2 ;; -1 ;; 2]]]]]; a[0] = 1; Array[a, 102, 0] (* Amiram Eldar, Jul 29 2025 *)

a(n)=my(r,k,t=n^3); for(i=0,exponent(t), if(bittest(t,i), k=0, k++>r, r=k)); if(n, r, 1) \\ Charles R Greathouse IV, Mar 10 2020

a(n) = A087117(A000578(n)).
a(n) <= 3*A070939(n) - 3 for n > 0. - Charles R Greathouse IV, Mar 10 2020
a(n) >= 3*A007814(n). Conjecture: if n < 2^k then a(n) < 3*k. - Robert Israel, Mar 10 2020

A090046 Length of longest contiguous block of 0's in binary expansion of n-th prime.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 3, 2, 1, 1, 0, 2, 2, 1, 1, 1, 1, 1, 4, 3, 2, 2, 2, 2, 4, 2, 2, 1, 1, 3, 0, 5, 3, 3, 2, 2, 2, 3, 2, 1, 2, 1, 1, 5, 3, 3, 2, 1, 3, 2, 2, 1, 3, 1, 7, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 1, 1, 4, 2, 1, 1, 1, 1, 4, 3, 3, 2, 3, 2, 1, 3, 1, 1, 5, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 5, 5, 4, 3, 3, 3, 3, 3

Offset: 1

Reinhard Zumkeller, Nov 20 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A004676, A023416, A087117, A090000, A090047, A090048, A090049.

f[n_] := Length[ Union[ DeleteCases[ Split[ IntegerDigits[n, 2]], 1, 2]][[ -1]]]; Table[ f[ Prime[n]], {n, 1, 105}] (* Robert G. Wilson v, Dec 03 2003 *)

a(n) = A087117(A000040(n)).

A144789 Consider the runs of 0's in the binary representation of n, each of these runs being on the edge of the binary representation n and/or being bounded by 1's. a(n) = the length of the shortest such run (with positive length) of 0's in binary n. a(n) = 0 if there are no runs of 0's in binary n.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 0, 3, 2, 1, 1, 2, 1, 1, 0, 4, 3, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 5, 4, 1, 3, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 0, 6, 5, 1, 4, 2, 1, 1, 3, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 1, 1, 2, 1, 1

Offset: 1

Leroy Quet, Sep 21 2008

			20 in binary is 10100. The runs of 0's are as follows: 1(0)1(00). The shortest of these runs contains exactly one 0's So a(20) = 1.

Cf. A087117, A144790.

A007814 := proc(n) local nshf,a ; a := 0 ; nshf := n ; while nshf mod 2 = 0 do nshf := nshf/2 ; a := a+1 ; od: a ; end: A144789 := proc(n) option remember ; local lp2,lp2sh,bind ; bind := convert(n,base,2) ; if add(i,i=bind) = nops(bind) then RETURN(0) ; fi; lp2 := A007814(n) ; if lp2 = 0 then A144789(floor(n/2)) ; else lp2sh := A144789(n/2^lp2) ; if lp2sh = 0 then lp2 ; else min(lp2,lp2sh) ; fi; fi; end: for n from 1 to 140 do printf("%d,",A144789(n)) ; od: # R. J. Mathar, Sep 29 2008

Table[Min[Length/@Select[Split[IntegerDigits[n,2]],MemberQ[#,0]&]],{n,120}]/.\[Infinity]->0 (* Harvey P. Dale, Jul 24 2016 *)

Extended by R. J. Mathar, Sep 29 2008

A330166 Length of the longest run of 0's in the ternary expression of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Joshua Oliver, Dec 04 2019

All numbers appear in this sequence. The n-th power of 3 (A000244(n)) has n 0's in its ternary expression.

The longest run of zeros possible in this sequence is 2, as the last digit of the ternary expression of the integers cycles between 0, 1, and 2, meaning that at least one of three consecutive numbers has a 0 in its ternary expression.

			For n = 87, the ternary expression of 87 is 10020. The length of the runs of 0's in the ternary expression of 87 are 2 and 1, respectively. The larger of these two values is 2, so a(87) = 2.
   n [ternary n] a(n)
   0 [        0] 1
   1 [        1] 0
   2 [        2] 0
   3 [      1 0] 1
   4 [      1 1] 0
   5 [      1 2] 0
   6 [      2 0] 1
   7 [      2 1] 0
   8 [      2 2] 0
   9 [    1 0 0] 2
  10 [    1 0 1] 1
  11 [    1 0 2] 1
  12 [    1 1 0] 1
  13 [    1 1 1] 0
  14 [    1 1 2] 0
  15 [    1 2 0] 1
  16 [    1 2 1] 0
  17 [    1 2 2] 0
  18 [    2 0 0] 2
  19 [    2 0 1] 1
  20 [    2 0 2] 1

Wikipedia, Ternary numeral system.

Cf. A000244, A007089, A077267, A087117, A330036, A330167, A330168.

Equals zero iff n is in A032924.

Table[Max@FoldList[If[#2==0,#1+1,0]&,0,IntegerDigits[n,3]],{n,0,90}]

a(A000244(n)) = a(3^n) = n.

a(n) = 0 iff n is in A032924.

A333767 Length of shortest run of zeros after a one in the binary expansion of n. a(0) = 0.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0

Offset: 0

Gus Wiseman, Apr 06 2020

nonn

			The binary expansion of 148 is (1,0,0,1,0,1,0,0), so a(148) = 1.

Positions of first appearances (ignoring index 0) are A000079.
Positions of terms > 0 are A022340.
Minimum prime index is A055396.
The  maximum part minus 1 is given by A087117.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without 1's are A022340.
- Sum is A070939.
- Product is A124758.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Maximum is A333766.
- Minimum is A333768.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A228351, A328594, A333217, A333218, A333219, A333632.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,Min@@stc[n]-1],{n,0,100}]

For n > 0, a(n) = A333768(n) - 1.

A218076 Indices of positive Fibonacci numbers whose binary expansions have record numbers of consecutive zeros.

Original entry on oeis.org

3, 6, 12, 19, 38, 42, 68, 243, 384, 515, 740, 1709, 5151, 11049, 45641, 94729, 185610, 644593, 726681, 2296396, 3098358, 6178778, 15743325, 22436908, 80141430, 84300971, 127495932, 177416979, 198423144, 275354607

Offset: 1

Peter Polm, Oct 20 2012

Positions of records in the auxiliary sequence 0, 0, 0, 1, 0, 1, 3, 1, 1, 3, 1, 2, 4, 2, 2, 3, 1, 3, 4, 5, 2,... = A087117(Fibonacci(n)). - R. J. Mathar, Nov 05 2012

			The first four records occur at 3, 6, 12, and 19:
F(3) = 10_2 (one zero),
F(6) = 1000_2 (three zeros),
F(12) = 10010000_2 (four zeros), and
F(19) = 1000001010101_2 (five zeros).
For the n=6178778, F(6178778) has 43 consecutive zeros.

Peter Polm, A218076 three more with C on 64 bits Linux (BigInteger Algorithms blog).
Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A059016, A087117.

More terms from Peter Polm's web site, Joerg Arndt, Aug 18 2014

cp's OEIS Frontend

A330036 The length of the largest run of 0's in the binary expansion of n + the length of the largest run of 1's in the binary expansion of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A090047 Length of longest contiguous block of 0's in binary expansion of n^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A090048 Length of longest contiguous block of 0's in binary expansion of n-th triangular number.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A090049 Length of longest contiguous block of 0's in binary expansion of n^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A090046 Length of longest contiguous block of 0's in binary expansion of n-th prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A330166 Length of the longest run of 0's in the ternary expression of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A333767 Length of shortest run of zeros after a one in the binary expansion of n. a(0) = 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula