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A254756 Numbers such that all their proper hexadecimal prefixes and suffixes represent primes.

%I A254756 #22 May 22 2025 10:21:42
%S A254756 34,35,37,39,43,45,50,51,53,55,59,61,82,83,85,87,91,93,114,115,117,
%T A254756 119,123,125,178,179,181,183,187,189,210,211,213,215,219,221,595,661,
%U A254756 663,669,691,693,763,851,947,949,979,1333,1339,1341,1429
%N A254756 Numbers such that all their proper hexadecimal prefixes and suffixes represent primes.
%C A254756 A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
%C A254756 Alternative definition: Slicing the hexadecimal expansion of a(n) in any way into two nonempty parts, each part represents a prime number.
%C A254756 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 A254756 The relation of a(n) to A237600 leads to the fastest way to reliably enumerate all its members.
%H A254756 Stanislav Sykora, <a href="/A254756/b254756.txt">Table of n, a(n) for n = 1..100</a>
%H A254756 Stanislav Sykora, <a href="https://oeis.org/wiki/File:GeneticThreads.txt">PARI/GP scripts for genetic threads</a>
%e A254756 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 A254756 1339, equal to 0x53B, is a member because all its proper hexadecimal prefixes and postfixes (0x5, 0x53, 0x3B, and 0xB) are prime.
%e A254756 The largest member is 0x259D397.
%o A254756 (PARI) \\ For the function GT_Trunc1 see A237600 and/or the link.
%o A254756 slicesIntoPrimes(n, b=10) = { \\ Same function as in A254751.
%o A254756 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 A254756 NumbersSlicingIntoPrimes(nmax,b=10) = {
%o A254756 my(rtp=GT_Trunc1(nmax,isprime,b)); \\ rtp right-truncatable primes
%o A254756 my(a=vector(b*#rtp),irtp,d,an,n=0);
%o A254756 for(irtp=1,#rtp, \\ For each rtp, append a digit and test
%o A254756    for(d=0,b-1,an=b*rtp[irtp]+d;
%o A254756      if(slicesIntoPrimes(an,b),n++;a[n]=an)););
%o A254756 return(a[1..n]);} v = NumbersSlicingIntoPrimes(1000000,16) \\ Call with nmax>>414,base 16
%o A254756 (Python)
%o A254756 from gmpy2 import is_prime
%o A254756 A254756_list = []
%o A254756 for n in range(16,10**6):
%o A254756     s = format(n,'x')
%o A254756     for i in range(1,len(s)):
%o A254756         if not (is_prime(int(s[i:],16)) and is_prime(int(s[:-i],16))):
%o A254756             break
%o A254756     else:
%o A254756         A254756_list.append(n) # _Chai Wah Wu_, Apr 16 2015
%Y A254756 Cf. A237600, A254751.
%K A254756 nonn,base,fini,full
%O A254756 1,1
%A A254756 _Stanislav Sykora_, Mar 05 2015