This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A124374 #6 Feb 16 2025 08:33:03
%S A124374 2,5,17,2957,23117,204557,2018957,4578979328975537786697650470157,
%T A124374 12572230784049013026617689884981971446439568309146114097251787122217783800812199225999909965168264460210470157
%N A124374 Primes of the form !(k + 1)/2 = Sum_{i=0..k} i!/2.
%C A124374 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 A124374 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha131.htm">Factorizations of many number sequences</a>.
%H A124374 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LeftFactorial.html">Left Factorial</a>.
%F A124374 a(n) = A003422(A124375(k) + 1)/2.
%t A124374 f=0;Do[f=f+n!;If[PrimeQ[f/2],Print[{n,f/2}]],{n,0,353}]
%Y A124374 Cf. A003422, A100614, A124375.
%K A124374 nonn
%O A124374 1,1
%A A124374 _Alexander Adamchuk_, Oct 28 2006