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

34, 35, 37, 39, 43, 45, 50, 51, 53, 55, 59, 61, 82, 83, 85, 87, 91, 93, 114, 115, 117, 119, 123, 125, 178, 179, 181, 183, 187, 189, 210, 211, 213, 215, 219, 221, 595, 661, 663, 669, 691, 693, 763, 851, 947, 949, 979, 1333, 1339, 1341, 1429

Offset: 1

Author

Stanislav Sykora, Mar 05 2015

Keywords

• nonn
• base
• fini
• full

Comments

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

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

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).

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

Examples

			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.
1339, equal to 0x53B, is a member because all its proper hexadecimal prefixes and postfixes (0x5, 0x53, 0x3B, and 0xB) are prime.
The largest member is 0x259D397.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..100
• Stanislav Sykora, PARI/GP scripts for genetic threads

Crossrefs

Cf. A237600, A254751.

Programs

PARI

\\ For the function GT_Trunc1 see A237600 and/or the link.
slicesIntoPrimes(n, b=10) = { \\ Same function as in A254751.
my(k=b); if(n0, if(!isprime(n\k)||!isprime(n%k), return(0); ); k*=b; ); return(1); }
NumbersSlicingIntoPrimes(nmax,b=10) = {
my(rtp=GT_Trunc1(nmax,isprime,b)); \\ rtp right-truncatable primes
my(a=vector(b*#rtp),irtp,d,an,n=0);
for(irtp=1,#rtp, \\ For each rtp, append a digit and test
   for(d=0,b-1,an=b*rtp[irtp]+d;
     if(slicesIntoPrimes(an,b),n++;a[n]=an)););
return(a[1..n]);} v = NumbersSlicingIntoPrimes(1000000,16) \\ Call with nmax>>414,base 16

Python

from gmpy2 import is_prime
A254756_list = []
for n in range(16,10**6):
    s = format(n,'x')
    for i in range(1,len(s)):
        if not (is_prime(int(s[i:],16)) and is_prime(int(s[:-i],16))):
            break
    else:
        A254756_list.append(n) # Chai Wah Wu, Apr 16 2015