A114337 Primes which are 1/3 of the cumulative sum of factorials of primes, if 1 is used as the zeroth prime.

3, 43, 1723, 13307323

Offset: 1

Author

Jonathan Vos Post, Feb 07 2006

Keywords

• fini
• full
• nonn

Comments

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

Examples

			prime(0)! = 1! = 1; prime(1)! = 2! = 2.
a(1) = (1! + 2! + 3!)/3 = 9/3 = 3.
a(2) = (1! + 2! + 3! + 5!)/3 = 129/3 = 43.
a(3) = (1! + 2! + 3! + 5! + 7!)/3 = 5169/3 = 1723.
a(4) = (1! + 2! + 3! + 5! + 7! + 11!)/3 = 39921969/3 = 13307323.
		

Crossrefs

Cf. A000040, A000142.

Programs

Mathematica

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

Formula

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.