A105184 Primes that can be written as concatenation of two primes in decimal representation.

23, 37, 53, 73, 113, 137, 173, 193, 197, 211, 223, 229, 233, 241, 271, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 433, 523, 541, 547, 571, 593, 613, 617, 673, 677, 719, 733, 743, 761, 773, 797, 977, 1013, 1033, 1093

Offset: 1

Author

Lekraj Beedassy, Apr 11 2005

Keywords

• nonn
• base

Comments

Primes that can be written as the concatenation of two distinct primes is the same sequence.

Number of terms < 10^n: 0, 4, 48, 340, 2563, 19019, 147249, ... - T. D. Noe, Oct 04 2010

The second prime cannot begin with the digit zero, else 307 would be the first additional term. - Michael S. Branicky, Sep 01 2024

Examples

			193 is in the sequence because it is the concatenation of the primes 19 and 3.
197 is in the sequence because it is the concatenation of the primes 19 and 7.
199 is not in the sequence because there is no way to break it into two substrings such that both are prime: neither 1 nor 99 is prime, and 19 is prime but 9 is not.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..10000

Crossrefs

Subsequence of A019549.

Cf. A121608, A121609, A121610, A083427, A129800.

Programs

Mathematica

searchMax = 10^4; Union[Reap[Do[p = Prime[i]; q = Prime[j]; n = FromDigits[Join[IntegerDigits[p], IntegerDigits[q]]]; If[PrimeQ[n], Sow[n]], {i, PrimePi[searchMax/10]}, {j, 2, PrimePi[searchMax/10^Ceiling[Log[10, Prime[i]]]]}]][[2, 1]]] (* T. D. Noe, Oct 04 2010 *)
Select[Prime@Range@1000,
 MatchQ[IntegerDigits@#, {x__, y__} /;
    PrimeQ@FromDigits@{x} && First@{y} != 0 &&
PrimeQ@FromDigits@{y}] &] (* Hans Rudolf Widmer, Nov 30 2024 *)

Python

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n)
    return any(s[i]!="0" and isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s)))
print([k for k in range(1100) if ok(k)]) # Michael S. Branicky, Sep 01 2024

Extensions

Corrected and extended by Ray Chandler, Apr 16 2005

Edited by N. J. A. Sloane, May 03 2007

Edited by N. J. A. Sloane, to remove erroneous b-file, comments and Mma program, Oct 04 2010