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

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

Author

Robert G. Wilson v, Dec 03 2003

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A090046, A090593.

Programs

Mathematica

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}]

Formula

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

Extensions

a(29)-a(31) from Donovan Johnson, Sep 10 2013