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

%I A080165 #22 Oct 25 2021 08:40:20
%S A080165 2,5,11,17,19,23,37,41,43,47,67,71,73,79,83,89,131,137,139,149,151,
%T A080165 157,163,167,173,179,181,191,257,263,269,271,277,281,283,293,307,311,
%U A080165 313,317,331,337,347,349,353,359,367,373,379,383,521,523,541,547,557,563
%N A080165 Primes having initial digits "10" in binary representation.
%C A080165 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
%C A080165 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
%H A080165 Alois P. Heinz, <a href="/A080165/b080165.txt">Table of n, a(n) for n = 1..20000</a>
%H A080165 G. Harman, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002426536">Primes in short intervals</a>, Math. Zeit., 180 (1982), 335-348.
%e A080165 A000040(15)=47 -> '101111' therefore 47 is a term.
%t A080165 Select[Prime[Range[1000]], IntegerDigits[#, 2][[;;2]] == {1, 0}&] (* _Jean-François Alcover_, Oct 25 2021 *)
%o A080165 (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" ")) ) }
%Y A080165 Cf. A004676, A080167.
%Y A080165 Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4, 5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
%Y A080165 Column k=2 of A262365.
%K A080165 nonn,base
%O A080165 1,1
%A A080165 _Reinhard Zumkeller_, Feb 03 2003