A254756 -id:A254756 - cp's OEIS frontend

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

Stanislav Sykora, Feb 15 2014

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

			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.

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

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

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

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)

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

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

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

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

A254751 Numbers such that, in base 10, all their proper prefixes and suffixes represent primes.

Original entry on oeis.org

22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 237, 297, 313, 317, 373, 537, 597, 713, 717, 737, 797, 2337, 2397, 2937, 3113, 3137, 3173, 3797, 5937, 5997, 7197, 7337, 7397, 29397, 31373, 37937, 59397, 73313, 739397

Offset: 1

Stanislav Sykora, Feb 15 2015

A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
Alternative definition: Slicing the decimal expansion of a(n) in any way into two nonempty parts, each part represents a prime number.
Every proper prefix of each member a(n) is a member of A024770, and every proper suffix is a member of A024785. Since these are finite sequences, a(n) is also finite. It has 45 members, the largest of which is 739397 and happens to be a prime.
The sequence is a union of A254753 and A020994.
A subsequence of A260181. - M. F. Hasler, Sep 16 2016

			6 is not a member because its expansion cannot be sliced in two.
597 is a member because (5,97,59, and 7) are all primes.
2331 is excluded because 233 is prime, but 1 is not. - _Gordon Hamilton_, Feb 20 2015

Cf. A020994, A024770, A024785, A254750, A254752, A254753, A254754, A254756.

Cf. A260181.

fQ[n_] := (p = {2, 3, 5, 7}; If[ Union@ Join[p, {Mod[n, 10]}] != p, {False}, Block[{idn = IntegerDigits@ n, lng = Floor@ Log10@ n}, Union@ PrimeQ@ Flatten@ Table[{FromDigits[ Take[idn, i]], FromDigits[ Take[idn, -lng + i - 1]]}, {i, lng}] == {True}]]); Select[ Range@1000000, fQ] (* Robert G. Wilson v, Feb 21 2015 *)
Select[Range[10,750000],AllTrue[Flatten[Table[FromDigits/@TakeDrop[IntegerDigits[#],n],{n,IntegerLength[#]-1}]],PrimeQ]&] (* Harvey P. Dale, Feb 13 2024 *)

slicesIntoPrimes(n,b=10) = {my(k=b);if(n0,if(!isprime(n\k)||!isprime(n%k),return(0););k*=b;);return(1);}

def breakIntoPrimes(n):
    D=n.digits()
    for i in [1..len(D)-1]:
        if not(is_prime(sum(D[i:][j]*10^j for j in range(len(D[i:])))) and is_prime(sum(D[:i][j]*10^j for j in range(len(D[:i]))))):
            return False
        else:
            continue
    return True
[n for n in [10..1000] if breakIntoPrimes(n)] # Tom Edgar, Feb 20 2015

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A237600 Right-truncatable primes in base 16.

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A254751 Numbers such that, in base 10, all their proper prefixes and suffixes represent primes.

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