A254756 - OEIS

%I #20 Apr 17 2015 05:27:42

%S 34,35,37,39,43,45,50,51,53,55,59,61,82,83,85,87,91,93,114,115,117,

%T 119,123,125,178,179,181,183,187,189,210,211,213,215,219,221,595,661,

%U 663,669,691,693,763,851,947,949,979,1333,1339,1341,1429

%N Numbers such that all their proper hexadecimal prefixes and suffixes represent primes.

%C A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.

%C Alternative definition: Slicing the hexadecimal expansion of a(n) in any way into two nonempty parts, each part represents a prime number.

%C Every proper hexadecimal prefix of each member a(n) must be a member of A237600. Since the latter is a finite sequence, a(n) is also finite. It has exactly 100 members, the largest of which is 39441303 (not a prime; the largest of the 16 primes occurring in this sequence is 3389).

%C The relation of a(n) to A237600 leads to the fastest way to reliably enumerate all its members.

%H Stanislav Sykora, <a href="/A254756/b254756.txt">Table of n, a(n) for n = 1..100</a>

%H Stanislav Sykora, <a href="https://oeis.org/wiki/File:GeneticThreads.txt">PARI/GP scripts for genetic threads</a>

%e 13 is not a member because its expansion in base 16 (0xD) cannot be sliced in two. 33 (equal to 0x21) is also not a member because 1 is not a prime, while 34 (equal to 0x22) is a member because 2 is a prime.

%e 1339, equal to 0x53B, is a member because all its proper hexadecimal prefixes and postfixes (0x5, 0x53, 0x3B, and 0xB) are prime.

%e The largest member is 0x259D397.

%o (PARI) \\ For the function GT_Trunc1 see A237600 and/or the link.

%o slicesIntoPrimes(n, b=10) = { \\ Same function as in A254751.

%o my(k=b); if(n<b, return(0); ); while(n\k>0, if(!isprime(n\k)||!isprime(n%k), return(0); ); k*=b; ); return(1); }

%o NumbersSlicingIntoPrimes(nmax,b=10) = {

%o my(rtp=GT_Trunc1(nmax,isprime,b)); \\ rtp right-truncatable primes

%o my(a=vector(b*#rtp),irtp,d,an,n=0);

%o for(irtp=1,#rtp, \\ For each rtp, append a digit and test

%o    for(d=0,b-1,an=b*rtp[irtp]+d;

%o      if(slicesIntoPrimes(an,b),n++;a[n]=an)););

%o return(a[1..n]);} v = NumbersSlicingIntoPrimes(1000000,16) \\ Call with nmax>>414,base 16

%o (Python)

%o from gmpy2 import is_prime

%o A254756_list = []

%o for n in range(16,10**6):

%o ....s = format(n,'x')

%o ....for i in range(1,len(s)):

%o ........if not (is_prime(int(s[i:],16)) and is_prime(int(s[:-i],16))):

%o ............break

%o ....else:

%o ........A254756_list.append(n) # _Chai Wah Wu_, Apr 16 2015

%Y Cf. A237600, A254751.

%K nonn,base,fini,full

%O 1,1

%A _Stanislav Sykora_, Mar 05 2015