A090000 -id:A090000 - cp's OEIS frontend

A038374 Length of longest contiguous block of 1's in binary expansion of n.

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

Offset: 1

Jeffrey Shallit

			a(157) = 3 because 157 in base 2 is 10011101 and longest contiguous block of 1's is of length 3.
May be arranged into blocks of lengths 1, 2, 4, 8, 16, ...:
1,
1, 2,
1, 1, 2, 3,
1, 1, 1, 2, 2, 2, 3, 4,
1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 2, 3, 3, 4, 5,
1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 4, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6,
... - _N. J. A. Sloane_, Jul 25 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A087117, A090000, A090001, A090002, A090003, A090050, A245196.

import Data.List (unfoldr, group)
a038374 = maximum . map length . filter ((== 1) . head) . group .
   unfoldr (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, May 01 2012

A038374 := proc(n) local nshft,thisr,resul; nshft := n ; resul :=0 ; thisr :=0 ; while nshft > 0 do if nshft mod 2 <> 0 then thisr := thisr+1 ; else resul := max(resul,thisr) ; thisr := 0 ; fi ; nshft := floor(nshft/2) ; od ; resul := max(resul,thisr) ; RETURN(resul) ; end : for n from 1 to 80 do printf("%d,",A038374(n)) ; od : # R. J. Mathar, Jun 15 2006

Table[Max[Length/@DeleteCases[Split[IntegerDigits[n,2]],?(MemberQ[ #,0] &)]],{n,120}] (* _Harvey P. Dale, Jun 10 2013 *)

a(n)=if (n==0, return (0)); n>>=valuation(n, 2); if(n<2, return(n)); my(e=valuation(n+1, 2)); max(e, a(n>>e)) \\ Charles R Greathouse IV, Jan 12 2014; edited by Michel Marcus, Apr 14 2019

from itertools import groupby
def a(n): return max(len(list(g)) for k, g in groupby(bin(n)[1:]) if k=='1')
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Jul 04 2022

a(n) >= A089309(n). a(n) >= A089310(n). a(2^i)=1. a(2^i-1)=i. - R. J. Mathar, Jun 15 2006

May be defined by the recurrence given in A245196, taking G(n)=n+1 (n>=0) and m=1. - N. J. A. Sloane, Jul 25 2014

A090001 Length of longest contiguous block of 1's in binary expansion of n^2.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Nov 20 2003

a(n) = A038374(A000290(n)).

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000120, A090003, A090002, A090000, A089998, A090047.

Join[{0},Table[Max[Length/@Select[Split[IntegerDigits[n^2,2]], MemberQ[ #,1]&]],{n,110}]] (* Harvey P. Dale, Nov 28 2014 *)

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)).

A090593 Smallest prime with exactly n consecutive ones in the longest run of ones in its binary expansion.

Original entry on oeis.org

2, 3, 7, 47, 31, 191, 127, 1021, 3583, 4093, 6143, 16381, 8191, 81919, 262139, 1114111, 131071, 786431, 524287, 4194301, 14680063, 16777213, 67108859, 654311423, 671088637, 738197503, 536870909, 5368709117, 3758096383, 34359738337, 2147483647, 21474836479

Offset: 1

Robert G. Wilson v, Dec 03 2003

			a(1) = 2 since 2_d = 10_b. a(2) = 3 since 3_d = 11_b. a(3) = 7 since 7_d = 111_b. a(4) = 47 since 47_d = 101111_b.

Chai Wah Wu, Table of n, a(n) for n = 1..3310

Cf. A090000, A090587.

a = Table[0, {30}]; NextPrim[n_] := Block[{k = n + 1}, While[ ! PrimeQ[k], k++ ]; k]; p = 2; Do[ m = Length[ Union[ DeleteCases[ Split[ IntegerDigits[p, 2]], 0, 2]][[ -1]]]; If[ a[[m]] == 0, a[[m]] = p]; p = NextPrim[p], {n, 1, 117000000}]

a(28)-a(32) from Donovan Johnson, Sep 10 2013

A307503 Least prime containing at least n consecutive 1's in its binary representation.

Original entry on oeis.org

2, 2, 3, 7, 31, 31, 127, 127, 1021, 3583, 4093, 6143, 8191, 8191, 81919, 131071, 131071, 131071, 524287, 524287, 4194301, 14680063, 16777213, 67108859, 536870909, 536870909, 536870909, 536870909, 2147483647, 2147483647, 2147483647, 2147483647, 21474836479

Offset: 0

John Mason, Apr 11 2019

For n > 0, a(n) = A000040(m) for the lowest m such that A090000(m) >= n.

a(n) = A087522(n) for n = 0 through 7, and in all other cases when a(n) is a base 2 repunit (Mersenne) prime.

			a(0) = 2, the smallest prime containing >= 0 1's.
a(1) = 2, the smallest prime containing >= 1 consecutive 1's.
a(2) = 3, the smallest prime containing >= 2 consecutive 1's.

Chai Wah Wu, Table of n, a(n) for n = 0..3314

Cf. A038374, A090000, A087522, A201914.

Cf. A090593 (with exactly n consecutive ones).

nbo(n)=if (n==0, return (0)); n>>=valuation(n, 2); if(n<2, return(n)); my(e=valuation(n+1, 2)); max(e, nbo(n>>e)); \\ A038374
a(n) = my(p=2); while(nbo(p) < n, p=nextprime(p+1)); p; \\ Michel Marcus, Apr 14 2019

a(n) <= A201914(n). - Rémy Sigrist, Apr 11 2019

a(n) = min_{k>=n} A090593(k). - Chai Wah Wu, Apr 26 2019

a(28)-a(32) from Chai Wah Wu, Apr 26 2019

cp's OEIS Frontend

A038374 Length of longest contiguous block of 1's in binary expansion of n.

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A090001 Length of longest contiguous block of 1's in binary expansion of n^2.

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

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A090593 Smallest prime with exactly n consecutive ones in the longest run of ones in its binary expansion.

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A307503 Least prime containing at least n consecutive 1's in its binary representation.

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