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Greatest 4th power integer that divides n!
3

%I #10 Sep 01 2024 09:37:35

%S 1,1,1,1,1,16,16,16,1296,20736,20736,20736,20736,20736,20736,331776,

%T 331776,429981696,429981696,268738560000,268738560000,268738560000,

%U 268738560000,4299816960000,4299816960000,4299816960000,348285173760000,13379723235164160000

%N Greatest 4th power integer that divides n!

%C Every term divides all its successors.

%H Clark Kimberling, <a href="/A248764/b248764.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = n!/A248766(n).

%F From _Amiram Eldar_, Sep 01 2024: (Start)

%F a(n) = A008835(n!).

%F a(n) = A248765(n)^4. (End)

%e a(6) = 16 because 16 divides 6! and if k > 2 then k^4 does not divide 6!.

%t z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];

%t u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];

%t v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];

%t p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];

%t m = 4; Table[p[m, n], {n, 1, z}] (* A248764 *)

%t Table[p[m, n]^(1/m), {n, 1, z}] (* A248765 *)

%t Table[n!/p[m, n], {n, 1, z}] (* A248766 *)

%t 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 (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 Cf. A000142, A008835, A248762, A248765, A248766.

%K nonn,easy

%O 1,6

%A _Clark Kimberling_, Oct 14 2014