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Search: id:A135291
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| A135291 |
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a(n) = the product of the non-zero exponents in the prime-factorization of n!. |
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+0
1
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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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COMMENT
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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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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 (ie. 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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A005361 := proc(n) mul( op(2, i), i=ifactors(n)[2]) ; end: A135291 := proc(n) A005361(n!) ; end: seq(A135291(n), n=0..50) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 12 2007
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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 (stefan.steinerberger(AT)gmail.com), Dec 05 2007
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CROSSREFS
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Cf. A005361.
Adjacent sequences: A135288 A135289 A135290 this_sequence A135292 A135293 A135294
Sequence in context: A105342 A021751 A093366 this_sequence A058617 A138135 A113166
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KEYWORD
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nonn
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AUTHOR
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Leroy Quet (qq-quet(AT)mindspring.com), Dec 03 2007
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EXTENSIONS
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More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com) and R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 05 2007
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