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

%I #5 Mar 21 2021 01:06:31

%S 2,5,17,2957,23117,204557,2018957,4578979328975537786697650470157,

%T 12572230784049013026617689884981971446439568309146114097251787122217783800812199225999909965168264460210470157

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

%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 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, ...}.

%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>.

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

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

%Y Cf. A003422, A100614, A124375.

%K nonn

%O 1,1

%A _Alexander Adamchuk_, Oct 28 2006