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A001222
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Number of prime divisors of n counted with multiplicity (also called bigomega(n) or Omega(n)).
(Formerly M0094 N0031)
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756
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0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 5, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 3, 1, 4, 3, 2, 1, 5, 1, 3, 2
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OFFSET
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1,4
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COMMENTS
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Maximal number of terms in any factorization of n.
Number of prime powers (not including 1) that divide n.
Sum of exponents in prime-power factorization of n. [From Daniel Forgues, Mar 29 2009]
Sum_{d|n} 2^(-A001221(d) - A001222(n/d)) = Sum_{d|n} 2^(-A001222(d) - A001221(n/d)) = 1 (see Dressler and van de Lune link). - Michel Marcus, Dec 18 2012
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 119, #12, omega(n)..
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.10.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 844.
B. Cloitre, A tauberian approach to RH, arXiv:1107.0812, 2011
Robert E. Dressler and Jan van de Lune, Some remarks concerning the number theoretic functions omega and Omega, Proc. Amer. Math. Soc. 41 (1973), 403-406
S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92.
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Roundness
Wolfram Research, First 50 numbers factored
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FORMULA
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n = Product (p_j^k_j) -> a(n) = Sum (k_j).
Dirichlet generating function: ppzeta(s)*zeta(s). Here ppzeta(s) = sum_{p prime} sum_{k=1}^{infinity} 1/(p^)k^s. Note that ppzeta(s) = sum_{p prime} 1/(p^s-1) and ppzeta(s) = sum_{k=1}^{infinity} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005.
Totally additive with a(p) = 1.
a(n) = if n=1 then 0 else a(n/A020639(n)) + 1. - Reinhard Zumkeller, Feb 25 2008
a(n) = sum(k=1..A001221(n), A124010(n,k) ). [Reinhard Zumkeller, Aug 27 2011]
a(n) = A022559(n)-A022559(n-1)
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EXAMPLE
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16=2^4, so a(16)=4; 18=2*3^2, so a(18)=3.
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MAPLE
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with(numtheory): seq(bigomega(n), n=1..111);
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MATHEMATICA
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Array[ Plus @@ Last /@ FactorInteger[ # ] &, 105]
PrimeOmega[Range[120]] (* From Harvey P. Dale, Apr 25 2011 *)
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PROG
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(PARI) vector(100, n, bigomega(n))
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CROSSREFS
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Cf. A001221 (primes counted without multiplicity), A046660, A144494. Bisections give A091304 and A073093. A086436 is essentially the same sequence.
a(n) = A091222(A091202(n)).
Sequences listing n such that a(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011
Sequence in context: A122810 A179953 A086436 * A098893 A069248 A008481
Adjacent sequences: A001219 A001220 A001221 * A001223 A001224 A001225
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KEYWORD
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nonn,easy,nice,core
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from David W. Wilson.
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STATUS
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approved
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