A090593 -id:A090593 - cp's OEIS frontend

A090000 Length of longest contiguous block of 1's in binary expansion of n-th prime.

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

Offset: 1

Reinhard Zumkeller, Nov 20 2003

a(n) = A038374(A000040(n)).

John Mason, Table of n, a(n) for n = 1..20000

Cf. A000120, A004676, A090001, A090002, A090003, A090046, A090593.

f[n_] := Length[ Union[ DeleteCases[ Split[ IntegerDigits[n, 2]], 0, 2]][[ -1]]]; Table[ f[ Prime[n]], {n, 1, 105}] (* Robert G. Wilson v, Dec 04 2003 *)

A090587 Smallest prime with exactly n consecutive zeros in the longest run of zeros in its binary expansion.

Original entry on oeis.org

3, 2, 19, 17, 67, 131, 641, 257, 2053, 10243, 4099, 12289, 40961, 32771, 65539, 65537, 262147, 786433, 4194319, 7340033, 23068673, 50331653, 67108879, 436207619, 167772161, 268435463, 268435459, 1073741831, 1073741827, 3221225473, 21474836483, 68719476767

Offset: 0

Robert G. Wilson v, Dec 03 2003

Except for 2, the first and last binary digits of a prime number are 1.

One may also define a sequence of the smallest prime with its longest run of zeros containing *at least* n zeros in the binary expansion: 2, 2, 17, 17, 67, 131, 257, 257, 2053, 4099,.... - R. J. Mathar, Sep 09 2013

			a(0) = 3 since 3_d = 11_b. a(1) = 2 since 2_d = 10_b. a(3) = 17 since 17_d = 10001_b. a(6) = 641 since 641_d = 1010000001_b.

Chai Wah Wu, Table of n, a(n) for n = 0..3313

Cf. A090046, A090593.

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]], 1, 2]][[ -1]]]; If[ a[[m + 1]] == 0, a[[m + 1]] = p]; p = NextPrim[p], {n, 1, 117000000}]

min{ A000040(k): A090046(k) = n}. - R. J. Mathar, Sep 09 2013

a(29)-a(31) 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

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A090000 Length of longest contiguous block of 1's in binary expansion of n-th prime.

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A090587 Smallest prime with exactly n consecutive zeros in the longest run of zeros 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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