%I #31 Aug 05 2021 02:25:46

%S 2,5,2,2,13,151,3,83,73,1433,647,29,28211,337,19,73,18181,130349,

%T 771079,731957,6619,4111,61927,140001721,42829,774885169,745984697,

%U 41711914513,34311919,117695654963,1139908799,2390249,54413,4707207067,129164452987,12496027

%N Largest prime dividing Sum_{k=0..n} k! * (n-k)!.

%C Sum_{k=0..n} k! * (n-k)! = (n+1)! * Sum_{k=0..n} 1 / ((k+1) * 2^(n-k)).

%H Sean A. Irvine, <a href="/A049413/b049413.txt">Table of n, a(n) for n = 1..315</a> (terms 1..167 from Robert Israel)

%e a(5)=13 because Sum_{k=0..5} k! * (5-k)! = 312 = 2^3*3*13.

%p for n from 1 to 33 do s := 0:for k from 0 to n do s := s+k!*(n-k)!:od: ifactor(s); od;

%t Table[FactorInteger[Sum[k!(n-k)!,{k,0,n}]][[-1,1]],{n,40}] (* _Harvey P. Dale_, May 23 2015 *)

%o (PARI) a(n) = my(f = factor(sum(k=0, n, k!*(n-k)!))); f[#f~,1]; \\ _Michel Marcus_, May 18 2014

%o (Python)

%o from sympy import factorial as f, primefactors

%o def a(n): return max(primefactors(sum(f(k)*f(n-k) for k in range(n+1))))

%o print([a(n) for n in range(1, 37)]) # _Michael S. Branicky_, Jul 31 2021

%K nonn

%O 1,1

%A _Leroy Quet_

%E More terms from Andrew Gacek (andrew(AT)dgi.net), Apr 21 2000

%E Corrected by _Jud McCranie_, Jan 03 2001

%E a(34)-a(36) from _Michel Marcus_, May 18 2014