A237600 -id:A237600 - cp's OEIS frontend

A238851 Right-truncatable, reversible primes in base 16.

2, 3, 5, 7, 11, 13, 53, 59, 61, 83, 89, 113, 179, 191, 211, 863, 947, 977, 983, 991, 1429, 1439, 1823, 3061, 3067, 3389, 15161, 15643, 15733, 15737, 15739, 15859, 23029, 48989, 48991, 251737, 251831, 253751, 368471, 4060019

Offset: 1

Stanislav Sykora, Mar 06 2014

See A238850 for definitions, and A238854 for comments on general context.

These numbers are fully right-truncatable and reversible primes in base 16 (but listed in decimal format). They are 40 in all.

			The largest such number (4060019) is in hex format 0x3DF373. It is a prime, so is 0x373FD3, and 0x3DF37 has again the same properties.

Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Cf. All in base 10: A238850, 100: A238852, 256: A238853.
Cf. In base n: A238854 (largest), A238855 (totals), A238856 (maximum digits), A238857 (m-digit counts).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

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

Original entry on oeis.org

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

Stanislav Sykora, Mar 05 2015

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.

			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.

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

Cf. A237600, A254751.

\\ 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

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

cp's OEIS Frontend

A238851 Right-truncatable, reversible primes in base 16.

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