login
a(n) is the next number after n! with the same number of prime factors as n!, counted with multiplicity.
0

%I #19 Sep 23 2024 09:27:07

%S 3,9,36,162,800,5248,41984,364544,3639168,39937536,479250432,

%T 6227066880,87178936320,1307674935296,20922798964736,355687506903040

%N a(n) is the next number after n! with the same number of prime factors as n!, counted with multiplicity.

%F A001222(a(n)) = A022559(n).

%F a(n+1) <= (n+1) * a(n), with equality for n = 2, 3, 7, 11, 13, 15, ...

%e a(4) = 36 because 4! = 24 = 2^3 * 3 and 36 = 2^2 * 3^2 both have 4 prime factors, counted with multiplicity, and no numbers between 24 and 36 have exactly 4 prime factors.

%p f:= proc(n) local x,t;

%p t:= numtheory:-bigomega(n);

%p for x from n!+1 do

%p if numtheory:-bigomega(x) = t then return x fi

%p od

%p end proc:

%p map(f, [$2..16]);

%t s={};Do[i=n!+1; ponf=PrimeOmega[n!];While[!ponf==PrimeOmega[i],i++];AppendTo[s,i] ,{n,2,14}];s (* _James C. McMahon_, Sep 23 2024 *)

%Y Cf. A000142, A001222, A022559.

%K nonn

%O 2,1

%A _Robert Israel_, Sep 22 2024