A129909 Primes that are the quotient of the decimal representation of concatenated twin primes divided by 3.

11, 19, 977, 1381, 1987, 75743, 93761, 115783, 213881, 273941, 285953, 4097077, 4337101, 4937161, 5737241, 6497317, 6757343, 8957563, 9097577, 10397707, 13057973, 14058073, 15158183, 15458213, 15998267, 17438411, 18338501

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..27.

Example

The first concatenated twin prime pair in decimal representation is 35.
The quotient of 35/3 is 11 which is prime and the first term.

Mathematica

Join[{11}, Select[FromDigits[Flatten[IntegerDigits/@#]]/3&/@ Select[ Partition[ Prime[Range[800]], 2, 1], #[[2]]-#[[1]]==2&], PrimeQ]] (* Harvey P. Dale, Aug 21 2021 *)

Prog

(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", ")) ) ) }

Crossrefs

Sequence in context: A295834 A129908 A344431 * A174976 A003284 A257401

Adjacent sequences: A129906 A129907 A129908 * A129910 A129911 A129912

Keyword

base,frac,nonn

Author

Cino Hilliard, Jun 05 2007

Status

approved