Search: id:A001222
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%I A001222 M0094 N0031
%S A001222 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,
%T A001222 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,
%U A001222 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
%N A001222 Number of prime divisors of n (counted with multiplicity).
%C A001222 Also called bigomega(n) or Omega(n).
%C A001222 Maximal number of terms in any factorization of n.
%C A001222 Number of prime powers (not including 1) that divide n.
%C A001222 Sum of exponents in prime-power factorization of n. [From Daniel Forgues 
               (squid(AT)zensearch.com), Mar 29 2009]
%D A001222 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.
%D A001222 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 119, #12, omega(n)..
%D A001222 M. Kac, Statistical Independence in Probability, Analysis and Number 
               Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
%D A001222 Amarnath Murthy, Generalization of Parition Function and Introducing 
               Smarandache Factor Partitions, Smarandache Notions Journal Vol. 11, 
               1-2-3 Spring 2000.
%D A001222 Amarnath Murthy, Length and Extent of Smarandache Factor Partitions, 
               Smarandache Notions Journal Vol. 11, 1-2-3 Spring 2000.
%D A001222 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.
%D A001222 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001222 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001222 Daniel Forgues, <a href="b001222.txt">Table of n, a(n) for n=1..100000</
               a>
%H A001222 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001222 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/
               ">Smarandache Notions Journal</a>
%H A001222 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/
               Cpaper35/page1.htm">The normal number of prime factors of a number</
               a>, Quart. J. Math. 48 (1917), 76-92.
%H A001222 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeFactor.html">Link to a section of The World of Mathematics.</
               a>
%H A001222 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Roundness.html">Link to a section of The World of Mathematics.</a>
%H A001222 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/
               FactorInteger/03/02">First 50 numbers factored</a>
%F A001222 n = Product (p_j^k_j) -> a(n) = Sum (k_j).
%F A001222 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.
%F A001222 Totally additive with a(p) = 1.
%F A001222 a(n) = if n=1 then 0 else a(n/A020639(n)) + 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 25 2008
%e A001222 16=2^4, so a(16)=4; 18=2*3^2, so a(18)=3.
%p A001222 with(numtheory): seq(bigomega(n),n=1..111);
%t A001222 Array[ Plus @@ Last /@ FactorInteger[ # ] &, 105]
%o A001222 (PARI) v=[ ]; for (n=1,100,v=concat(v,bigomega(n))); v
%Y A001222 Cf. A001221 (primes counted without multiplicity), A046660, A144494. 
               Bisections give A091304 and A073093. A086436 is essentially the same 
               sequence.
%Y A001222 a(n) = A091222(A091202(n)).
%Y A001222 Sequence in context: A116479 A122810 A086436 this_sequence A098893 A069248 
               A008481
%Y A001222 Adjacent sequences: A001219 A001220 A001221 this_sequence A001223 A001224 
               A001225
%K A001222 nonn,easy,nice,core
%O A001222 1,4
%A A001222 N. J. A. Sloane (njas(AT)research.att.com).
%E A001222 More terms from David W. Wilson (davidwwilson(AT)comcast.net).

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