A237600 Right-truncatable primes in base 16.

2, 3, 5, 7, 11, 13, 37, 41, 43, 47, 53, 59, 61, 83, 89, 113, 127, 179, 181, 191, 211, 223, 593, 599, 601, 607, 659, 661, 691, 701, 757, 761, 853, 857, 859, 863, 947, 953, 977, 983, 991, 1427, 1429, 1433, 1439, 1811, 1823, 2039, 2879, 2897, 2903, 2909, 3061

Offset: 1

Author

Stanislav Sykora, Feb 15 2014

Keywords

• nonn
• base
• fini
• full
• easy
• tabf

Comments

Numbers with these properties: (i) a(n) is a prime and (ii) its image under the function E(k) = k\16 = floor(k/16) is zero or has the same properties. [Corrected by M. F. Hasler, Nov 07 2018]

The sequence has 414 nonzero members.

Otherwise said, integers p > 0 such that floor(p/16^k) is prime or zero for all k >= 0. One might relax to p >= 0, i.e., include an initial term 0, corresponding to an empty string of digits. The recursive definition can also be used to produce all of the terms, starting with the primes < 16, and adding, for each term of the list, the primes made from appending a digit to that term, i.e., the primes between 16 x that term and 16 more. The sequence can also be seen as a table whose n-th row yields the terms with n digits in base 16: row lengths are A237601 and the last term of row n is A237602(n). - M. F. Hasler, Nov 07 2018

Examples

			a(414) = 16778492037124607, in hexadecimal notation 3B9BF319BD51FF, belongs to a(n) because each of its hexadecimal prefixes (including itself) is a prime. Being the largest of such numbers, it is also a member of A023107.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..414
• Stanislav Sykora, PARI/GP scripts for genetic threads

Crossrefs

Cf. A023107, A024770 (base 10), A237601, A237602, A254756.

Programs

Mathematica

Select[Range@ 3600, AllTrue[Most[DeleteDuplicates@ FixedPointList[f, #]], PrimeQ] &] (* Michael De Vlieger, Mar 07 2015, Version 10 *)

PARI

GT_Trunc1(nmax,prop,b=10) = { \\ See the link for details
  my (n=0,v=vector(nmax),g=1,lgs=1,lge,an,c);
  for (k=1,b-1,if (prop(k),v[n++]=k));
  lge=n; c=lge-lgs+1;
  while (c, g++;for (k=lgs,lge,for (m=0,b-1, an=b*v[k]+m;
    if (prop(an), v[n++]=an;if (n>=nmax,return (v)));););
    lgs=lge+1; lge=n; c=lge-lgs+1;);
  if (n, return (v[1..n]));
  print("No solution");}
v = GT_Trunc1(1000000,isprime,16)

PARI

isok(n)={ while(n, if(!isprime(n),return(0));n\=16); 1} \\ Joerg Arndt, Mar 07 2015

PARI

my(A=primes([0,15]),i=1); until(#AA237600=A \\ M. F. Hasler, Nov 07 2018

Python

from gmpy2 import is_prime
A237600_list = []
for n in range(1,10**9):
    if is_prime(n):
        s = format(n,'x')
        for i in range(1,len(s)):
            if not is_prime(int(s[:-i],16)):
                break
        else:
            A237600_list.append(n) # Chai Wah Wu, Apr 16 2015

Python

from sympy import primerange
p = lambda x: list(primerange(x,x+16)); A237600 = p(0); i=0
while iA237600): A237600+=p(A237600[i]*16); i+=1 # M. F. Hasler, Mar 11 2020