|
%I #9 Mar 21 2021 01:06:37
%S 2,3,4,7,8,9,10,29,75,162,270,272,353,720,1795,3732,4768,9315,12220,
%T 41531
%N Numbers k such that A003422(k+1)/2 is prime.
%C 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, ...}.
%C A near-duplicate of A100614: a(n) = A100614(n) - 1. - _Ryan Propper_, Feb 07 2008
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha131.htm">Factorizations of many number sequences</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeftFactorial.html">Left Factorial</a>.
%t f=0;Do[f=f+n!;If[PrimeQ[f/2],Print[{n,f/2}]],{n,0,353}]
%t Flatten[Position[Accumulate[(Range[0,12220]!)]/2,_?PrimeQ]]-1 (* _Harvey P. Dale_, Jul 02 2019 *)
%Y Cf. A003422, A124374.
%K nonn,more,hard
%O 1,1
%A _Alexander Adamchuk_, Oct 28 2006
%E More terms from _Ryan Propper_, Feb 07 2008
%E a(20) from _Jinyuan Wang_, Mar 20 2021
|
