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Primes which are 1/3 of the cumulative sum of factorials of primes, if 1 is used as the zeroth prime.
1

%I #12 Aug 01 2015 11:52:58

%S 3,43,1723,13307323

%N Primes which are 1/3 of the cumulative sum of factorials of primes, if 1 is used as the zeroth prime.

%C No more primes. Starting with a(14) = (1! + ... + 43!)/3 the sum always has a factor of 47.

%F Defining prime(0)= 1: a(n) = (1/3)*Sum_{i=0..n}A000142(A000040(i+1)) iff in A000040. a(n) = (1/3)*Sum_{i=0..n}prime(i+1)! iff in A000040.

%e prime(0)! = 1! = 1; prime(1)! = 2! = 2.

%e a(1) = (1! + 2! + 3!)/3 = 9/3 = 3.

%e a(2) = (1! + 2! + 3! + 5!)/3 = 129/3 = 43.

%e a(3) = (1! + 2! + 3! + 5! + 7!)/3 = 5169/3 = 1723.

%e a(4) = (1! + 2! + 3! + 5! + 7! + 11!)/3 = 39921969/3 = 13307323.

%t f[n_] := (1 + Plus @@ ((Prime@ Range@ n)!))/3; Select[f /@ Range@ 43, PrimeQ@# &] (* _Robert G. Wilson v_, Apr 30 2009 *)

%Y Cf. A000040, A000142.

%K fini,full,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 07 2006