%I #69 Nov 04 2022 19:53:30

%S 3,5,23,47,89,101,149,157,163,173,179,199,229,313,331,367,379,383,443,

%T 457,523,587,631,643,647,653,659,709,883,947,997,1009,1091,1097,1163,

%U 1259,1277,1283,1289,1321,1483,1601,1669,1693,1709,1753,1877,2063,2069,2099

%N Primes whose binary representation is also the decimal representation of a prime.

%C In general rebase notation (Marc LeBrun): p2 = (2) [p] (10).

%C Also: Primes in A036952. - _M. F. Hasler_, Dec 11 2012

%C See A089971 for the binary representation of these terms. - _M. F. Hasler_, Jan 05 2014

%H Harry J. Smith and K. D. Bajpai, <a href="/A065720/b065720.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Harry J. Smith)

%H M. F. Hasler, <a href="https://docs.google.com/document/d/10IM7fcAbB2tqRGuwfGvuEGUzD_IXbgXPDK0tfxN4M3o/pub">Primes whose base c expansion is also the base b expansion of a prime</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_280.htm">Puzzle 280. 3893257</a>, The Prime Puzzles & Problems Connection.

%F Equals A036952 intersect A000040. - _M. F. Hasler_, Dec 11 2012

%e 1009{10} = 1111110001{2} is prime, and 1111110001{10} is also prime.

%e 89 is in the sequence because it is a prime. Binary representation of 89 = 1011001, which is also a prime.

%p select(t -> isprime(t) and isprime(convert(t,binary)),[seq(2*i+1,i=1..1000)]); # _Robert Israel_, Jul 08 2014

%t Select[ Range[1900], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 2]]] & ]

%t Select[ Prime@ Range@ 330, PrimeQ[ FromDigits[ IntegerDigits[#, 2]]] &] (* _Robert G. Wilson v_, Oct 09 2014 *)

%o (PARI) {(baseE(x, b)= local(d, e=0, f=1); while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); e); n=0; for (m=1, 10^9, p=prime(m); b=baseE(p, 2); if (isprime(b), write("b065720.txt", n++, " ", p); if (n==1000, return)) ) } \\ _Harry J. Smith_, Oct 27 2009

%o (PARI) isok(p) = isprime(p) && isprime(fromdigits(binary(p), 10)); \\ _Michel Marcus_, Mar 04 2022

%o (Python)

%o from sympy import isprime

%o def ok(n): return isprime(n) and isprime(int(bin(n)[2:]))

%o print([k for k in range(2100) if ok(k)]) # _Michael S. Branicky_, Mar 04 2022

%Y Cf. A065721, A065722, A065723, A065724, A065725, A065726, A065727, A065361.

%Y Cf. A123266, A256621, A256622.

%K nonn,base

%O 1,1

%A _Patrick De Geest_, Nov 15 2001

%E a(48)-a(50) from _K. D. Bajpai_, Jul 04 2014