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%I #7 Mar 07 2019 19:44:26
%S 43,157,19183,22651,37057,121453,7923366007441921,
%T 4496830293424385744456428801,45045561823582321,412807,
%U 49907098805169447878401,34672666242568358583785606401,1041421
%N Primes of the form (p! + q!)/ p! where p= prime(k) and q= prime(k+1), in order of increasing k.
%C The 6th term has 6 digits; the 44th term has 44 digits.
%C The 685th term has 349 digits.
%H K. D. Bajpai, <a href="/A235392/b235392.txt">Table of n, a(n) for n = 1..1000</a>
%e 43 is in the sequence because (5! + 7!)/ 5! = (120 + 5040)/120 = 43 which is prime and 5 and 7 are consecutive primes.
%e 157 is in the sequence because (11! + 13!)/ 11! = (39916800 + 6227020800)/ 39916800 = 157 which is prime and 11 and 13 are consecutive primes.
%p KD := proc() local a,b,d; a:=ithprime(n); b:=ithprime(n+1); d:=(a! + b!)/ a!; if isprime(d) then RETURN (d); fi; end: seq(KD(), n=1..300);
%t Select[((#[[1]]!+#[[2]]!)/#[[1]]!&/@Partition[Prime[Range[ 300]], 2,1]), PrimeQ] (* _Harvey P. Dale_, Mar 07 2019 *)
%Y Cf. A000040 (prime numbers).
%Y Cf. A100858 (primes:(p-1)! + p).
%K nonn
%O 1,1
%A _K. D. Bajpai_, Jan 09 2014
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