A124374 Primes of the form !(k + 1)/2 = Sum_{i=0..k} i!/2.

2, 5, 17, 2957, 23117, 204557, 2018957, 4578979328975537786697650470157, 12572230784049013026617689884981971446439568309146114097251787122217783800812199225999909965168264460210470157

Offset: 1

Author

Alexander Adamchuk, Oct 28 2006

Keywords

• nonn

Comments

Sum_{i=0..k} i! = k! + !k = A003422(k+1), where !k is left factorial !k = Sum_{i=0..k-1} i! = A003422(k). Left factorials are even for k > 1. Corresponding numbers k such that Sum_{i=0..k} i!/2 = A003422(k+1)/2 is prime are listed in A124375(n) = {2, 3, 4, 7, 8, 9, 10, 29, 75, 162, 270, 272, 353, ...}.

Links

• Hisanori Mishima, Factorizations of many number sequences.
• Eric Weisstein's World of Mathematics, Left Factorial.

Crossrefs

Cf. A003422, A100614, A124375.

Programs

Mathematica

f=0;Do[f=f+n!;If[PrimeQ[f/2],Print[{n,f/2}]],{n,0,353}]

Formula

a(n) = A003422(A124375(k) + 1)/2.