A124375 Numbers k such that A003422(k+1)/2 is prime.

2, 3, 4, 7, 8, 9, 10, 29, 75, 162, 270, 272, 353, 720, 1795, 3732, 4768, 9315, 12220, 41531

Offset: 1

Author

Alexander Adamchuk, Oct 28 2006

Keywords

• nonn
• more
• hard

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 primes of the form (k! + !k)/2 = (a(n)! + !a(n))/2 are listed in A124374(n) = {2, 5, 17, 2957, 23117, 204557, 2018957, 4578979328975537786697650470157, ...}.

A near-duplicate of A100614: a(n) = A100614(n) - 1. - Ryan Propper, Feb 07 2008

Links

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

Crossrefs

Cf. A003422, A124374.

Programs

Mathematica

f=0;Do[f=f+n!;If[PrimeQ[f/2],Print[{n,f/2}]],{n,0,353}]
Flatten[Position[Accumulate[(Range[0,12220]!)]/2,?PrimeQ]]-1 (* _Harvey P. Dale, Jul 02 2019 *)

Extensions

More terms from Ryan Propper, Feb 07 2008

a(20) from Jinyuan Wang, Mar 20 2021