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A248764 Greatest 4th power integer that divides n!

%I A248764 #10 Sep 01 2024 09:37:35
%S A248764 1,1,1,1,1,16,16,16,1296,20736,20736,20736,20736,20736,20736,331776,
%T A248764 331776,429981696,429981696,268738560000,268738560000,268738560000,
%U A248764 268738560000,4299816960000,4299816960000,4299816960000,348285173760000,13379723235164160000
%N A248764 Greatest 4th power integer that divides n!
%C A248764 Every term divides all its successors.
%H A248764 Clark Kimberling, <a href="/A248764/b248764.txt">Table of n, a(n) for n = 1..1000</a>
%F A248764 a(n) = n!/A248766(n).
%F A248764 From _Amiram Eldar_, Sep 01 2024: (Start)
%F A248764 a(n) = A008835(n!).
%F A248764 a(n) = A248765(n)^4. (End)
%e A248764 a(6) = 16 because 16 divides 6! and if k > 2 then k^4 does not divide 6!.
%t A248764 z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
%t A248764 u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
%t A248764 v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
%t A248764 p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];
%t A248764 m = 4; Table[p[m, n], {n, 1, z}]  (* A248764 *)
%t A248764 Table[p[m, n]^(1/m), {n, 1, z}]   (* A248765 *)
%t A248764 Table[n!/p[m, n], {n, 1, z}]      (* A248766 *)
%t A248764 f[p_, e_] := p^(4*Floor[e/4]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30] (* _Amiram Eldar_, Sep 01 2024 *)
%o A248764 (PARI) a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(4*(f[i, 2]\4)));} \\ _Amiram Eldar_, Sep 01 2024
%Y A248764 Cf. A000142, A008835, A248762, A248765, A248766.
%K A248764 nonn,easy
%O A248764 1,6
%A A248764 _Clark Kimberling_, Oct 14 2014