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A022559
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Sum of exponents in prime-power factorization of n!.
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19
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0, 0, 1, 2, 4, 5, 7, 8, 11, 13, 15, 16, 19, 20, 22, 24, 28, 29, 32, 33, 36, 38, 40, 41, 45, 47, 49, 52, 55, 56, 59, 60, 65, 67, 69, 71, 75, 76, 78, 80, 84, 85, 88, 89, 92, 95, 97, 98, 103, 105, 108, 110, 113, 114, 118, 120, 124, 126, 128, 129, 133, 134, 136, 139
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OFFSET
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0,4
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COMMENTS
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For n > 0, partial sums of bigomega (A001222): number of prime divisors (counted with multiplicity). Subsequence of primes: 2, 5, 7, 11, 13, 19, 29, 41, 47, 59, 67, 71, 89, 97, 103, 113, 139, 151, 179, 191, 211, 229, 239, 251, 263, .... [Jonathan Vos Post, Feb 10 2010]
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LINKS
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Daniel Forgues, Table of n, a(n) for n = 0..100000
M. Hassani, On the decomposition of n! into primes, arXiv:math/0606316
K. Matthews, Computing the prime-power factorization of n!
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FORMULA
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a(n) = a(n-1)+A001222(n).
A027746(a(A000040(n))+1) = A000040(n). A082288(a(n)+1) = n.
A001221(n!) = omega(n!) = pi(n) = A000720(n).
a(n) = SUM[i=1..n] A001222(i) = SUM[i=1..n] bigomega(i) = SUM[i=1..n] {Number of prime divisors of i (counted with multiplicity)}. [Jonathan Vos Post, Feb 10 2010]
a(n) = n log log n + B_2 * n + o(n), with B_2 = A083342. [Charles R Greathouse IV, Jan 11 2012]
a(n) = A210241(n) - n for n > 0. [Reinhard Zumkeller, Mar 23 2012]
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MAPLE
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with(numtheory):with(combinat):a:=proc(n) if n=0 then 0 else bigomega(numbperm(n)) fi end: seq(a(n), n=0..63); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 11 2008
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MATHEMATICA
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Array[Plus@@Last/@FactorInteger[ #! ]&, 5!, 0] [From Vladimir Joseph Stephan Orlovsky, Nov 10 2009]
f[n_] := If[n <= 1, 0, Total[FactorInteger[n]][[2]]]; Accumulate[Array[f, 100, 0]] (* T. D. Noe, Apr 11 2011 *)
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PROG
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(PARI) a(n)=bigomega(n!)
(PARI) k=0; vector(n, i, k+=bigomega(i))
(Haskell)
a022559 n = a022559_list !! n
a022559_list = scanl (+) 0 $ map a001222 [1..]
-- Renhard Zumkeller, Feb 16 2012
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CROSSREFS
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Cf. A001222, A013939, A046660, A144494, A115627.
Sequence in context: A111040 A191324 A147807 * A049781 A174090 A076697
Adjacent sequences: A022556 A022557 A022558 * A022560 A022561 A022562
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KEYWORD
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nonn,nice,changed
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AUTHOR
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Karen E. Wandel (kw29(AT)evansville.edu)
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EXTENSIONS
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Typo corrected by Daniel Forgues, Nov 16 2009
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STATUS
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approved
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