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A135291
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Product of the nonzero exponents in the prime factorization of n!.
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8
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1, 1, 1, 1, 3, 3, 8, 8, 14, 28, 64, 64, 100, 100, 220, 396, 540, 540, 768, 768, 1152, 1944, 4104, 4104, 5280, 7920, 16560, 21528, 31200, 31200, 40768, 40768, 48608, 78120, 161280, 230400, 277440, 277440, 571200, 907200, 1108080, 1108080, 1440504
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OFFSET
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0,5
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COMMENTS
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a(n) = A005361(n!). For n >= 2, a(n) = the number of positive divisors of n! which themselves are each divisible by every prime <= n. For p = any prime, a(p) = a(p-1). a(0)=a(1)=1 because the product of the exponents is over the empty set.
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LINKS
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FORMULA
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EXAMPLE
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6! = 720 has a prime factorization of 2^4 * 3^2 * 5^1. So a(6) = 4*2*1 = 8.
Also, 720 is divisible by a(6)=8 positive divisors which themselves are each divisible by every prime <= 6 (i.e., are each divisible by 2*3*5 = 30): 30, 60, 90, 120, 180, 240, 360, 720.
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MAPLE
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MATHEMATICA
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Table[Product[FactorInteger[n! ][[i, 2]], {i, 1, Length[FactorInteger[n! ]]}], {n, 0, 50}] (* Stefan Steinerberger, Dec 05 2007 *)
Table[Times@@Transpose[FactorInteger[n!]][[2]], {n, 0, 50}] (* Harvey P. Dale, Aug 16 2011 *)
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PROG
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(PARI) valp(n, p)=my(s); while(n\=p, s+=n); s
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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