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A135354 a(0)=1, a(n) = largest divisor of n! that is coprime to a(n-1).

%I A135354 #16 Jul 22 2024 15:25:28
%S A135354 1,1,2,3,8,15,16,315,128,2835,256,155925,1024,6081075,2048,638512875,
%T A135354 32768,10854718875,65536,1856156927625,262144,194896477400625,524288,
%U A135354 49308808782358125,4194304,3698160658676859375,8388608,1298054391195577640625,33554432,263505041412702261046875,67108864
%N A135354 a(0)=1, a(n) = largest divisor of n! that is coprime to a(n-1).
%H A135354 Robert Israel, <a href="/A135354/b135354.txt">Table of n, a(n) for n = 0..504</a>
%F A135354 a(2n) = the largest power of 2 that divides (2n)!. a(2n+1) = the largest odd divisor of (2n+1)! = (2n+1)!/a(2n).
%p A135354 f:= proc(n,a)
%p A135354   local P,R,i;
%p A135354   P:= select(t -> isprime(t) and igcd(t,a)=1, [2,seq(i,i=3..n,2)]);
%p A135354   R:= map(proc(p) local k; add(floor(n/p^k), k=1 ..ilog[p](n)) end proc, P);
%p A135354   mul(P[i]^R[i],i=1..nops(P));
%p A135354 end proc:
%p A135354 R:= 1: r:= 1: for i from 1 to 50 do r:= f(i,r); R:= R,r od:
%p A135354 R; # _Robert Israel_, Jul 21 2024
%t A135354 a = {1}; For[n = 1, n < 25, n++, AppendTo[a, Select[Divisors[n! ], GCD[a[[ -1]], # ] == 1 &][[ -1]]]]; a (* _Stefan Steinerberger_, Dec 10 2007 *)
%t A135354 ldnf[{n_,a_}]:={n+1,Max[Select[Divisors[(n+1)!],CoprimeQ[#,a]&]]}; Transpose[ NestList[ldnf,{0,1},30]][[2]] (* _Harvey P. Dale_, Jan 21 2016 *)
%Y A135354 Cf. A060818, A049606.
%K A135354 nonn,look
%O A135354 0,3
%A A135354 _Leroy Quet_, Dec 07 2007
%E A135354 More terms from _Stefan Steinerberger_, Dec 10 2007
%E A135354 More terms from _Robert Israel_, Jul 21 2024