

A254756


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


3



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
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OFFSET

1,1


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.


LINKS

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


EXAMPLE

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.


PROG

(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(n<b, return(0); ); while(n\k>0, 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 righttruncatable 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, b1, 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


CROSSREFS

Cf. A237600, A254751.
Sequence in context: A291512 A165855 A271984 * A203462 A270311 A257315
Adjacent sequences: A254753 A254754 A254755 * A254757 A254758 A254759


KEYWORD

nonn,base,fini,full


AUTHOR

Stanislav Sykora, Mar 05 2015


STATUS

approved



