A080165 Primes having initial digits "10" in binary representation.

2, 5, 11, 17, 19, 23, 37, 41, 43, 47, 67, 71, 73, 79, 83, 89, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 521, 523, 541, 547, 557, 563

Offset: 1

Author

Reinhard Zumkeller, Feb 03 2003

Keywords

• nonn
• base

Comments

Also primes that terminate at 4,2,1 in the x-1 problem: Repeat, if x is even divide by 2 else subtract 1, until 4 is reached. - Cino Hilliard, Mar 27 2003

David W. Wilson remarks that it follows from standard results about primes in short intervals (see for example Harman, 1982) that there are infinitely many numbers in any base b starting with any nonzero prefix c. - N. J. A. Sloane, Sep 19 2015

Examples

			A000040(15)=47 -> '101111' therefore 47 is a term.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• G. Harman, Primes in short intervals, Math. Zeit., 180 (1982), 335-348.

Crossrefs

Cf. A004676, A080167.

Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4, 5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.

Column k=2 of A262365.

Programs

Mathematica

Select[Prime[Range[1000]], IntegerDigits[#, 2][[;;2]] == {1, 0}&] (* Jean-François Alcover, Oct 25 2021 *)

PARI

pxnm1(n,p) = { forprime(x=2,n, p1 = x; while(p1>1, if(p1%2==0,p1/=2,p1 = p1*p-1;); if(p1 == 4,break); ); if(p1 == 4,print1(x" ")) ) }