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%I #25 Feb 28 2023 14:21:28
%S 23,37,53,73,1117,1123,1129,1153,1171,1319,1361,1367,1373,1723,1741,
%T 1747,1753,1759,1783,1789,1913,1931,1973,1979,1997,2311,2341,2347,
%U 2371,2383,2389,2917,2953,2971,3119,3137,3167,3719,3761,3767,3779,3797,4111,4129,4153,4159,4337,4373,4397,4723,4729
%N Primes that are the concatenation of two primes with the same number of digits.
%C It appears that there are ~ 0.81*100^k/(k^2 log^2 10) 2k-digit numbers in this sequence, making their relative density 0.9/(k^2 log^2 10) among 2k-digit numbers. Of course there are no 2k+1-digit terms in the sequence. - _Charles R Greathouse IV_, Nov 15 2022
%C The second Mathematica program below generates all 2-digit and 4-digit terms of the sequence. To generate all 2,753 6-digit terms of the sequence, use this Mathematica program: Select[1000#[[1]]+#[[2]]&/@Tuples[Prime[Range[26,168]],2],PrimeQ]. There are 112,649 8-digit terms of the sequence. - _Harvey P. Dale_, Feb 28 2023
%H Robert Israel, <a href="/A358421/b358421.txt">Table of n, a(n) for n = 1..10000</a>
%e a(5) = 1117 is a term because 11 and 17 are both 2-digit primes and 1117 is prime.
%p Res:= NULL: count:= 0:
%p for d from 2 by 2 while count < 100 do
%p pq:= 10^(d-1);
%p while count < 100 do
%p pq:= nextprime(pq);
%p if pq > 10^d then break fi;
%p q:= pq mod 10^(d/2);
%p if q < 10^(d/2-1) then next fi;
%p p:= (pq-q)/10^(d/2);
%p if isprime(p) and isprime(q) then Res:= Res,pq; count:= count+1 fi
%p od od:
%p Res;
%t Select[Prime[Range[640]], EvenQ[(s = IntegerLength[#])] && IntegerDigits[#][[s/2 + 1]] > 0 && And @@ PrimeQ[QuotientRemainder[#, 10^(s/2)]] &] (* _Amiram Eldar_, Nov 15 2022 *)
%t Join[Select[FromDigits/@Tuples[Prime[Range[4]],2],PrimeQ],Select[100#[[1]]+ #[[2]]&/@ Tuples[Prime[Range[5,25]],2],PrimeQ]] (* _Harvey P. Dale_, Feb 28 2023 *)
%o (Python)
%o from itertools import count, islice
%o from sympy import isprime, primerange
%o def agen(): # generator of terms
%o for d in count(1):
%o pow = 10**d
%o for p in primerange(10**(d-1), pow):
%o for q in primerange(10**(d-1), pow):
%o t = p*pow + q
%o if isprime(t): yield t
%o print(list(islice(agen(), 51))) # _Michael S. Branicky_, Nov 15 2022
%Y Subsequence of A000040 and of A001637.
%K nonn,base,easy
%O 1,1
%A _J. M. Bergot_ and _Robert Israel_, Nov 15 2022
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