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A248762
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Greatest cube that divides n!.
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5
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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
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OFFSET
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1,4
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COMMENTS
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Every term divides all its successors.
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LINKS
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FORMULA
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EXAMPLE
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a(4) = 8 because 8 divides 24 and if k > 2 then k^3 does not divide 24.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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