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A133924
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a(n) = number of exponents occurring only once each in the prime factorization of n!.
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3
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0, 0, 1, 0, 2, 1, 3, 2, 2, 2, 4, 3, 3, 3, 2, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 6, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 4, 6, 6, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9
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OFFSET
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0,5
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LINKS
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FORMULA
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EXAMPLE
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14! is factored into primes as 2^11 * 3^5 * 5^2 * 7^2 * 11^1 * 13^1. The exponent 1 and 2 each occur more than once. So the exponents occurring only once each are 5 and 11. Therefore a(14) = 2.
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MAPLE
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A133924 := proc(n) local ifs, a, i ; if n <= 1 then RETURN(0) ; else ifs := ifactors(n!)[2] ; ifs := sort([seq(op(2, i), i=ifs)]) ; a :=0 ; for i from 1 to nops(ifs) do if i = 1 or op(i, ifs) <> op(i-1, ifs) then if i=nops(ifs) or op(i, ifs) <> op(i+1, ifs) then a := a+1 ; fi ; fi ; od: RETURN(a) ; fi ; end: seq(A133924(n), n=0..120) ; # R. J. Mathar, Jan 30 2008
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MATHEMATICA
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ne1[n_]:=Count[Tally[Transpose[FactorInteger[n!]][[2]]], _?(Last[#] == 1&)]; Join[{0, 0}, Array[ne1, 110, 2]] (* Harvey P. Dale, Aug 21 2011 *)
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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