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%I #4 Aug 21 2021 17:10:22
%S 11,19,977,1381,1987,75743,93761,115783,213881,273941,285953,4097077,
%T 4337101,4937161,5737241,6497317,6757343,8957563,9097577,10397707,
%U 13057973,14058073,15158183,15458213,15998267,17438411,18338501
%N Primes that are the quotient of the decimal representation of concatenated twin primes divided by 3.
%C Except for the first term, concatenated twin primes are always divisible by 3. This follows from the fact that twin prime components > 3 are of the form 6k-1 and 6k+1. So concatenation in decimal is (6k-1)*10^d + 6k+1 = 6k(10^d+1)+(10^d-1) where d is the number of digits in each twin prime component. Now 10^d-1 = (10-1)(10^(d-1)+10^(d-2)+...+1) = 9h and 6k(10^d+1) + 9h is divided by 3.
%e The first concatenated twin prime pair in decimal representation is 35.
%e The quotient of 35/3 is 11 which is prime and the first term.
%t Join[{11},Select[FromDigits[Flatten[IntegerDigits/@#]]/3&/@ Select[ Partition[ Prime[Range[800]],2,1],#[[2]]-#[[1]]==2&],PrimeQ]] (* _Harvey P. Dale_, Aug 21 2021 *)
%o (PARI) concattwins3p(n) = { local(x,y); forprime(x=2,n, if(isprime(x+2), y=eval(concat(Str(x),Str(x+2)))/3; if(isprime(y),print1(y",")) ) ) }
%K base,frac,nonn
%O 1,1
%A _Cino Hilliard_, Jun 05 2007
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