A248766 - OEIS

%I #11 Sep 01 2024 09:37:43

%S 1,2,6,24,120,45,315,2520,280,175,1925,23100,300300,4204200,63063000,

%T 63063000,1072071000,14889875,282907625,9053044,190113924,4182506328,

%U 96197645544,144296468316,3607411707900,93792704405400,31264234801800,22787343150,660832951350

%N Greatest 4th-power-free divisor of n!

%H Clark Kimberling, <a href="/A248766/b248766.txt">Table of n, a(n) for n = 1..1000</a>

%H Rafael Jakimczuk, <a href="https://doi.org/10.12988/imf.2017.7542">On the h-th free part of the factorial</a>, International Mathematical Forum, Vol. 12, No. 13 (2017), pp. 629-634.

%F a(n) = n!/A248764(n).

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

%F a(n) = A053165(n!).

%F log(a(n)) = 2*log(2) * n + o(n) (Jakimczuk, 2017). (End)

%e a(6) = 45 because 45 divides 6! and if k > 45 divides 6!, then h^4 divides 6!/k for some h > 1.

%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^Mod[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]^(f[i, 2] % 4)); \\ _Amiram Eldar_, Sep 01 2024

%Y Cf. A000142, A053165, A248764, A248765.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Oct 14 2014