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

%I A105184 #29 Dec 03 2024 12:45:12
%S A105184 23,37,53,73,113,137,173,193,197,211,223,229,233,241,271,283,293,311,
%T A105184 313,317,331,337,347,353,359,367,373,379,383,389,397,433,523,541,547,
%U A105184 571,593,613,617,673,677,719,733,743,761,773,797,977,1013,1033,1093
%N A105184 Primes that can be written as concatenation of two primes in decimal representation.
%C A105184 Primes that can be written as the concatenation of two distinct primes is the same sequence.
%C A105184 Number of terms < 10^n: 0, 4, 48, 340, 2563, 19019, 147249, ... - _T. D. Noe_, Oct 04 2010
%C A105184 The second prime cannot begin with the digit zero, else 307 would be the first additional term. - _Michael S. Branicky_, Sep 01 2024
%H A105184 T. D. Noe, <a href="/A105184/b105184.txt">Table of n, a(n) for n=1..10000</a>
%e A105184 193 is in the sequence because it is the concatenation of the primes 19 and 3.
%e A105184 197 is in the sequence because it is the concatenation of the primes 19 and 7.
%e A105184 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.
%t A105184 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 *)
%t A105184 Select[Prime@Range@1000,
%t A105184  MatchQ[IntegerDigits@#, {x__, y__} /;
%t A105184     PrimeQ@FromDigits@{x} && First@{y} != 0 &&
%t A105184 PrimeQ@FromDigits@{y}] &] (* _Hans Rudolf Widmer_, Nov 30 2024 *)
%o A105184 (Python)
%o A105184 from sympy import isprime
%o A105184 def ok(n):
%o A105184     if not isprime(n): return False
%o A105184     s = str(n)
%o A105184     return any(s[i]!="0" and isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s)))
%o A105184 print([k for k in range(1100) if ok(k)]) # _Michael S. Branicky_, Sep 01 2024
%Y A105184 Subsequence of A019549.
%Y A105184 Cf. A121608, A121609, A121610, A083427, A129800.
%K A105184 nonn,base
%O A105184 1,1
%A A105184 _Lekraj Beedassy_, Apr 11 2005
%E A105184 Corrected and extended by _Ray Chandler_, Apr 16 2005
%E A105184 Edited by _N. J. A. Sloane_, May 03 2007
%E A105184 Edited by _N. J. A. Sloane_, to remove erroneous b-file, comments and Mma program, Oct 04 2010