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A129908 Quotient of the decimal representation of concatenated twin primes divided by 3. 1


%S 11,19,371,573,977,1381,1987,2391,33701,35703,45713,49717,59727,63731,

%T 65733,75743,79747,89757,93761,103771,115783,139807,143811,153821,

%U 173841,189857,199867,205873,213881,219887,269937,273941,275943,285953

%N 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.

%H Harvey P. Dale, <a href="/A129908/b129908.txt">Table of n, a(n) for n = 1..1000</a>

%e The first concatenated twin prime pair in decimal representation is 35.

%e The quotient of 35/3 is 11 which is the first term.

%t Join[{11},FromDigits[Flatten[IntegerDigits/@#]]/3&/@Rest[Select[ Partition[ Prime[ Range[200]],2,1],Last[#]-First[#]==2&]]] (* _Harvey P. Dale_, Oct 12 2012 *)

%o (PARI) concattwins3(n) = { local(x,y); forprime(x=2,n, if(isprime(x+2), y=eval(concat(Str(x),Str(x+2)))/3; print1(y",")) ) }

%K base,frac,nonn

%O 1,1

%A _Cino Hilliard_, Jun 05 2007

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Last modified December 11 01:07 EST 2019. Contains 329910 sequences. (Running on oeis4.)