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

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

Author

Stanislav Sykora, Feb 15 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 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

Examples

			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
		

Crossrefs

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

Cf. A260181.

Programs

Mathematica

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 *)

PARI

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

Sage

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