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A248766
Greatest 4th-power-free divisor of n!
4
1, 2, 6, 24, 120, 45, 315, 2520, 280, 175, 1925, 23100, 300300, 4204200, 63063000, 63063000, 1072071000, 14889875, 282907625, 9053044, 190113924, 4182506328, 96197645544, 144296468316, 3607411707900, 93792704405400, 31264234801800, 22787343150, 660832951350
OFFSET
1,2
LINKS
Rafael Jakimczuk, On the h-th free part of the factorial, International Mathematical Forum, Vol. 12, No. 13 (2017), pp. 629-634.
FORMULA
a(n) = n!/A248764(n).
From Amiram Eldar, Sep 01 2024: (Start)
a(n) = A053165(n!).
log(a(n)) = 2*log(2) * n + o(n) (Jakimczuk, 2017). (End)
EXAMPLE
a(6) = 45 because 45 divides 6! and if k > 45 divides 6!, then h^4 divides 6!/k for some h > 1.
MATHEMATICA
z = 40; f[n_] := f[n] = FactorInteger[n!]; r[m_, x_] := r[m, x] = m*Floor[x/m];
u[n_] := Table[f[n][[i, 1]], {i, 1, Length[f[n]]}];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
p[m_, n_] := p[m, n] = Product[u[n][[i]]^r[m, v[n]][[i]], {i, 1, Length[f[n]]}];
m = 4; Table[p[m, n], {n, 1, z}] (* A248764 *)
Table[p[m, n]^(1/m), {n, 1, z}] (* A248765 *)
Table[n!/p[m, n], {n, 1, z}] (* A248766 *)
f[p_, e_] := p^Mod[e, 4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30] (* Amiram Eldar, Sep 01 2024 *)
PROG
(PARI) a(n) = my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(f[i, 2] % 4)); \\ Amiram Eldar, Sep 01 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 14 2014
STATUS
approved