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A248780
Number of cubes that divide n!
5
1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 8, 8, 8, 24, 36, 36, 36, 36, 42, 112, 112, 112, 128, 192, 192, 240, 270, 270, 270, 270, 330, 792, 792, 792, 864, 864, 864, 2016, 2912, 2912, 4704, 4704, 4704, 5376, 5760, 5760, 6144, 6144, 7680, 15360, 16320, 16320, 18360
OFFSET
1,4
LINKS
FORMULA
a(n) = product_{i=1..r} 1+floor(e[i]/3), where product_{i=1..r} p[i]^e[i] is the prime factorization of n!. - M. F. Hasler, Oct 22 2014
a(n) = A061704(A000142(n)). - Michel Marcus, Mar 27 2015
EXAMPLE
a(9) counts these divisors of 9!: 1, 8, 27, 64, 216, 1728.
MATHEMATICA
z = 130; m = 3; f[n_] := f[n] = FactorInteger[n!];
v[n_] := Table[f[n][[i, 2]], {i, 1, Length[f[n]]}];
a[n_] := Apply[Times, 1 + Floor[v[n]/m]]
A248780 = Table[a[n], {n, 1, z}] (* simplified by M. F. Hasler, Oct 22 2014 *)
PROG
(PARI) a(n)=sumdiv(n!, d, ispower(d, 3))
for(n=1, 50, print1(a(n), ", ")) \\ Derek Orr, Oct 20 2014, simplified by M. F. Hasler, Oct 22 2014
(PARI) A248780(n)=prod(i=1, #n=factor(n!)[, 2], 1+n[i]\3) \\ M. F. Hasler, Oct 22 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 15 2014
STATUS
approved