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A248762
Greatest cube that divides n!.
6
1, 1, 1, 8, 8, 8, 8, 64, 1728, 1728, 1728, 13824, 13824, 13824, 46656000, 2985984000, 2985984000, 2985984000, 2985984000, 23887872000, 221225582592000, 221225582592000, 221225582592000, 1769804660736000, 221225582592000000, 221225582592000000
OFFSET
1,4
COMMENTS
Every term divides all its successors.
LINKS
FORMULA
a(n) = n!/A145642(n).
From Amiram Eldar, Sep 01 2024: (Start)
a(n) = A008834(n!).
a(n) = A248763(n)^3. (End)
EXAMPLE
a(4) = 8 because 8 divides 24 and if k > 2 then k^3 does not divide 24.
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 = 3; Table[p[m, n], {n, 1, z}] (* A248762 *)
Table[p[m, n]^(1/m), {n, 1, z}] (* A248763 *)
Table[n!/p[m, n], {n, 1, z}] (* A145642 *)
gk[n_]:=Select[Divisors[n!], IntegerQ[Surd[#, 3]]&]; Max[#]&/@Array[gk, 30] (* Harvey P. Dale, Sep 16 2021 *)
f[p_, e_] := p^(3*Floor[e/3]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 30] (* Amiram Eldar, Sep 01 2024 *)
PROG
(PARI) a(n)=k=ceil((n!/2)^(1/3)); while(n!%k^3, k--); k^3
vector(20, n, a(n)) \\ Derek Orr, Oct 19 2014
(PARI) a(n) = {my(f = factor(n!)); prod(i = 1, #f~, f[i, 1]^(3*(f[i, 2]\3))); } \\ Amiram Eldar, Sep 01 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 14 2014
STATUS
approved