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A248824
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Number of integers k^6 that divide 1!*2!*3!*...*n!.
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1
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1, 1, 1, 1, 2, 3, 6, 8, 18, 42, 64, 200, 432, 546, 960, 3888, 6000, 15180, 29952, 38976, 67200, 285600, 393984, 1632960, 3175200, 4165392, 6105600, 38413440, 55339200, 114048000, 205632000, 280219500, 448156800, 2621445120, 3777725952, 12940849152
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OFFSET
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1,5
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LINKS
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EXAMPLE
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a(7) counts these integers k^6 that divide 125411328000 = A000178(6): 1, 64, 729, 4096, 46656, 2985984, these being k^6 for k = 1, 2, 3, 4, 6, 12.
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MAPLE
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b:= proc(n) option remember; add(i[2]*x^numtheory[pi](i[1]),
i=ifactors(n)[2])+`if`(n=1, 0, b(n-1))
end:
c:= proc(n) option remember; b(n)+`if`(n=1, 0, c(n-1)) end:
a:= n->(p->mul(iquo(coeff(p, x, i), 6)+1, i=1..degree(p)))(c(n)):
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MATHEMATICA
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z = 40; p[n_] := Product[k!, {k, 1, n}];
f[n_] := f[n] = FactorInteger[p[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]]}];
t[m_, n_] := Apply[Times, 1 + r[m, v[n]]/m]
m = 6; Table[t[m, n], {n, 1, z}] (* A248824 *)
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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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