%I A001221 M0056 N0019
%S A001221 0,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,3,1,1,2,2,2,
               2,1,
%T A001221 2,2,2,1,3,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,3,1,2,2,1,2,3,1,2,2,3,1,2,
               1,2,
%U A001221 2,2,2,3,1,2,1,2,1,3,2,2,2,2,1,3,2,2,2,2,2,2,1,2,2,2,1,3,1,2,3,2,1,2,1,
               3,2
%N A001221 Number of distinct primes dividing n (also called omega(n)).
%C A001221 Comments from Peter C. Heinig (algorithms(AT)gmx.de), Mar 08 2008: (Start) 
               This is also the number of maximal ideals of the ring (Z/nZ,+,*). 
               Since every finite integral domain must be a field, every prime ideal 
               of Z/nZ is a maximal ideal and since in general each maximal ideal 
               is prime, there are just as many prime ideals as maximal ones in 
               Z/nZ, so the sequence gives the number of prime ideals of Z/nZ as 
               well.
%C A001221 The reason why this number is given by the sequence is that the ideals 
               of Z/nZ are precisely the subgroups of (Z/nZ,+). Hence for an ideal 
               to be maximal it has form a maximal subgroup of (Z/nZ,+) and this 
               is equivalent to having prime index in (Z/nZ) and this is equivalent 
               to being generated by a single prime divisor of n.
%C A001221 Finally, all the groups arising in this way have different orders, hence 
               are different, so the number of maximal ideals equals the number 
               of distinct primes dividing n. (End)
%C A001221 Equals double inverse Mobius transform of A143519, where A051731 = the 
               inverse Mobius transform. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 22 2008]
%C A001221 a(n) = number of prime divisors of n. a(n) = number of prime-power divisors 
               of n. If n = Product (p_i^e_i), the prime-power divisors of n are 
               p_1^e_1, p_2^e_2, ..., p_k^e_k, where k = number of distinct primes 
               dividing n. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               May 04 2009]
%C A001221 sum_{k=0;inf} 1 / (10 ^ A000040(k) - 1) (see A073668) [From Eric Desbiaux 
               (moongerms(AT)wanadoo.fr), Jun 24 2009]
%D A001221 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 A001221 M. Kac, Statistical Independence in Probability, Analysis and Number 
               Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
%D A001221 J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) 
               (British Association Mathematical Tables Vol.V), Burlington House/
               Cambridge University Press London 1935.
%D A001221 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001221 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001221 Daniel Forgues, <a href="b001221.txt">Table of n, a(n) for n=1..100000</
               a>
%H A001221 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 A001221 H. Bottomley, <a href="http://www.btinternet.com/~se16/js/factor.htm">
               prime factors calculator</a>
%H A001221 J. Brennen, <a href="http://www.brennen.net/primes/FactorApplet.html">
               Prime Factoring Applet</a>
%H A001221 J. Britton, <a href="http://britton.disted.camosun.bc.ca/jbprimefactor.htm">
               Prime Factorization Machine</a>
%H A001221 A. Dendane, <a href="http://www.analyzemath.com/Calculators_3/prime_factors.html">
               Prime Factors Calculator</a>
%H A001221 J. Flament, <a href="http://jocelyn.smoofy.net/np/decomposition.php">
               Decomposition d'un nombre en facteurs premiers</a>
%H A001221 A. Hodges, <a href="http://www.cryptographic.co.uk/factorise.html">Java 
               Applet for Factorisation</a>
%H A001221 S. O. S. Math, <a href="http://www.sosmath.com/tables/factor/factor.html">
               Prime factorization of the first 1000 integers</a>
%H A001221 K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorization 
               and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)</a>
%H A001221 J. Moyer, <a href="http://www.rsok.com/~jrm/factor.html">"Prime Factors 
               of Integers" server for numbers up to 10^36</a>
%H A001221 Primefan, <a href="http://primefan.tripod.com/500factored.html">The First 
               2500 Integers,Factored</a>
%H A001221 Primefan, <a href="http://primefan.tripod.com/Factorer.html">Factorer</
               a>
%H A001221 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 A001221 F. Richman, <a href="http://www.math.fau.edu/Richman/mla/factor-f.htm">
               Factoring into Primes</a>
%H A001221 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DistinctPrimeFactors.html">Link to a section of The World of Mathematics.</
               a>
%H A001221 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MoebiusTransform.html">Link to a section of The World of Mathematics.</
               a>
%H A001221 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 A001221 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PrimeZetaFunction.html">Prime zeta function primezeta(s)</a>.
%H A001221 Wikipedia, <a href="http://en.wikipedia.org/wiki/Table_of_prime_factors">
               Table of prime factors</a>
%H A001221 D. Williams, <a href="http://www.louisville.edu/~dawill03/crypto/Factoring.html">
               Factoring</a>
%H A001221 G. Xiao, WIMS server, <a href="http://wims.unice.fr/~wims/en_tool~algebra~factor.en.html">
               Factoris</a>
%F A001221 G.f.: sum(k>=1, x^prime(k)/(1-x^prime(k))) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 21 2003
%F A001221 G.f.: sum(i=1, oo, isprime(i)/(1-x^i)) = sum(i=1, oo, isprime(i)*x^i/
               (1-x^i)), where isprime(n) returns 1 is n is prime, 0 otherwise. 
               - Jon Perry (perry(AT)globalnet.co.uk), Jul 03 2004
%F A001221 Dirichlet generating function: zeta(s)*primezeta(s). - Franklin T. Adams-Watters, 
               Sep 11 2005.
%F A001221 Additive with a(p^e) = 1.
%F A001221 a(1) = 0, a(p) = 1, a(pq) = 2, a(pq...z) = k, a(p^k) = 1, for p = primes 
               (A000040), pq = product of two distinct primes (A006881), pq...z 
               = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k 
               = prime powers (A000961(n) for n > 1), k = natural numbers (A000027). 
               [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), May 04 2009]
%p A001221 A001221 := proc(n) local t1, i; if n = 1 then RETURN(0) else t1 := 0; 
               for i to n do if n mod ithprime(i) = 0 then t1 := t1 + 1 end if end 
               do end if; t1 end proc;
%p A001221 with(numtheory): seq(nops(factorset(n)),n=1..120); (Deutsch)
%t A001221 Array[ Length[ FactorInteger[ # ] ]&, 100 ]
%o A001221 (MuPAD) func(nops(numlib::primedivisors(n)), n):
%o A001221 (PARI) a(n)=omega(n)
%o A001221 (MuPad) numlib::omega(n)$ n=1..110 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 13 2008
%Y A001221 Cf. A001222 (primes counted with multiplicity), A046660. Partial sums 
               give A013939.
%Y A001221 a(n) = A091221(A091202(n)).
%Y A001221 Cf. A087624, A143519, A144494.
%Y A001221 A156542. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 10 2009]
%Y A001221 Sequence in context: A158210 A087802 A079553 this_sequence A064372 A096825 
               A007875
%Y A001221 Adjacent sequences: A001218 A001219 A001220 this_sequence A001222 A001223 
               A001224
%K A001221 nonn,easy,nice,core
%O A001221 1,6
%A A001221 N. J. A. Sloane (njas(AT)research.att.com).
%E A001221 G.f. corrected by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Sep 01 2009
%E A001221 Replaced two geocities.com URL's - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 30 2009