A105184 - OEIS

%I #29 Dec 03 2024 12:45:12

%S 23,37,53,73,113,137,173,193,197,211,223,229,233,241,271,283,293,311,

%T 313,317,331,337,347,353,359,367,373,379,383,389,397,433,523,541,547,

%U 571,593,613,617,673,677,719,733,743,761,773,797,977,1013,1033,1093

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

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

%C Number of terms < 10^n: 0, 4, 48, 340, 2563, 19019, 147249, ... - _T. D. Noe_, Oct 04 2010

%C The second prime cannot begin with the digit zero, else 307 would be the first additional term. - _Michael S. Branicky_, Sep 01 2024

%H T. D. Noe, <a href="/A105184/b105184.txt">Table of n, a(n) for n=1..10000</a>

%e 193 is in the sequence because it is the concatenation of the primes 19 and 3.

%e 197 is in the sequence because it is the concatenation of the primes 19 and 7.

%e 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 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 Select[Prime@Range@1000,

%t  MatchQ[IntegerDigits@#, {x__, y__} /;

%t     PrimeQ@FromDigits@{x} && First@{y} != 0 &&

%t PrimeQ@FromDigits@{y}] &] (* _Hans Rudolf Widmer_, Nov 30 2024 *)

%o (Python)

%o from sympy import isprime

%o def ok(n):

%o     if not isprime(n): return False

%o     s = str(n)

%o     return any(s[i]!="0" and isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s)))

%o print([k for k in range(1100) if ok(k)]) # _Michael S. Branicky_, Sep 01 2024

%Y Subsequence of A019549.

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

%K nonn,base

%O 1,1

%A _Lekraj Beedassy_, Apr 11 2005

%E Corrected and extended by _Ray Chandler_, Apr 16 2005

%E Edited by _N. J. A. Sloane_, May 03 2007

%E Edited by _N. J. A. Sloane_, to remove erroneous b-file, comments and Mma program, Oct 04 2010