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A001221 Number of distinct primes dividing n (also called omega(n)).
(Formerly M0056 N0019)
683
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

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.

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.

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)

Equals double inverse Mobius transform of A143519, where A051731 = the inverse Mobius transform. [From Gary W. Adamson, Aug 22 2008]

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. [Jaroslav Krizek, May 04 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

Up to 2*3*5*7*11*13*17*19*23*29  - 1 = 6469693230 - 1, also the decimal expansion of the constant 0.01111211... = sum_{k>=0} 1 / (10 ^ A000040(k) - 1) (see A073668). [Eric Desbiaux, Jan 20 2014]

REFERENCES

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.

M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.

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.

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).

LINKS

N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 from NJAS)

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].

H. Bottomley, Prime factors calculator

J. Brennen, Prime Factoring Applet

J. Britton, Prime Factorization Machine

A. Dendane, Prime Factors Calculator

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

J. Flament, Decomposition d'un nombre en facteurs premiers

A. Hodges, Java Applet for Factorisation

S. O. S. Math, Prime factorization of the first 1000 integers

K. Matthews, Factorization and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)

J. Moyer, "Prime Factors of Integers" server for numbers up to 10^36

Primefan, The First 2500 Integers,Factored

Primefan, Factorer

S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92.

F. Richman, Factoring into Primes

Eric Weisstein's World of Mathematics, Distinct Prime Factors

Eric Weisstein's World of Mathematics, Moebius Transform

Eric Weisstein's World of Mathematics, Prime Factor

Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).

Wikipedia, Prime factor

Wikipedia, Table of prime factors

D. Williams, Factoring

G. Xiao, WIMS server, Factoris

FORMULA

G.f.: sum(k>=1, x^prime(k)/(1-x^prime(k))) - Benoit Cloitre, Apr 21 2003

G.f.: sum(i=1, infinity, isprime(i)/(1-x^i)) = sum(i=1, infinity, isprime(i)*x^i/(1-x^i)), where isprime(n) returns 1 is n is prime, 0 otherwise. - Jon Perry, Jul 03 2004

Dirichlet generating function: zeta(s)*primezeta(s). - Franklin T. Adams-Watters, Sep 11 2005.

Additive with a(p^e) = 1.

a(1) = 0, a(p) = 1, a(pq) = 2, a(pq...z) = k, a(p^k) = 1, where p, q,...,z are k distinct primes and k natural numbers. [Jaroslav Krizek, May 04 2009]

a(n) = log_2(Sum(d|n, mu(d)^2)). - Enrique Pérez Herrero, Jul 09 2012

MAPLE

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;

A001221 := proc(n) nops(numtheory[factorset](n)) end proc: # Emeric Deutsch

MATHEMATICA

Array[ Length[ FactorInteger[ # ] ]&, 100 ]

PrimeNu[Range[120]]  (* Harvey P. Dale, Apr 26 2011 *)

PROG

(MuPAD) func(nops(numlib::primedivisors(n)), n):

(MuPad) numlib::omega(n)$ n=1..110 - Zerinvary Lajos, May 13 2008

(PARI) a(n)=omega(n)

(Sage)

def A001221(n) : return len(filter(is_prime, divisors(n)))

[A001221(n) for n in (1..80)] # Peter Luschny, Feb 01 2012

CROSSREFS

Cf. A001222 (primes counted with multiplicity), A046660. Partial sums give A013939.

a(n) = A091221(A091202(n)).

Cf. A087624, A143519, A144494.

Cf. A156542. [From Reinhard Zumkeller, Feb 10 2009]

Sequence in context: A158210 A087802 A079553 * A064372 A096825 A007875

Adjacent sequences:  A001218 A001219 A001220 * A001222 A001223 A001224

KEYWORD

nonn,easy,nice,core

AUTHOR

N. J. A. Sloane.

EXTENSIONS

G.f. corrected by Franklin T. Adams-Watters, Sep 01 2009

STATUS

approved

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Last modified August 29 08:09 EDT 2014. Contains 246187 sequences.