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Search: id:A001222
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| A001222 |
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Number of prime divisors of n (counted with multiplicity).
(Formerly M0094 N0031)
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+0
469
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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
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Also called bigomega(n) or Omega(n).
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 (squid(AT)zensearch.com), Mar 29 2009]
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
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, Generalization of Parition Function and Introducing Smarandache Factor Partitions, Smarandache Notions Journal Vol. 11, 1-2-3 Spring 2000.
Amarnath Murthy, Length and Extent of Smarandache Factor Partitions, Smarandache Notions Journal Vol. 11, 1-2-3 Spring 2000.
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.
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LINKS
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Daniel Forgues, Table of n, a(n) for n=1..100000
N. J. A. Sloane, First 10000 values of Omega(n): Table of n, a(n) for n = 1..10000
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].
M. L. Perez et al., eds., Smarandache Notions Journal
S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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 (reinhard.zumkeller(AT)gmail.com), Feb 25 2008
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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]
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PROGRAM
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(PARI) v=[ ]; for (n=1, 100, v=concat(v, bigomega(n))); v
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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)).
Adjacent sequences: A001219 A001220 A001221 this_sequence A001223 A001224 A001225
Sequence in context: A116479 A122810 A086436 this_sequence A098893 A069248 A008481
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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 (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David W. Wilson (davidwwilson(AT)comcast.net).
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