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A001222 Number of prime divisors of n counted with multiplicity (also called bigomega(n) or Omega(n)).
(Formerly M0094 N0031)
835
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; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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. - 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

Row sums in A067255. - Reinhard Zumkeller, Jun 11 2013

REFERENCES

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.

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 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

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.

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

FORMULA

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).

EXAMPLE

16=2^4, so a(16)=4; 18=2*3^2, so a(18)=3.

MAPLE

with(numtheory): seq(bigomega(n), n=1..111);

MATHEMATICA

Array[ Plus @@ Last /@ FactorInteger[ # ] &, 105]

PrimeOmega[Range[120]] (* Harvey P. Dale, Apr 25 2011 *)

PROG

(PARI) vector(100, n, bigomega(n))

(MAGMA) [n eq 1 select 0 else &+[p[2]: p in Factorization(n)]: n in [1..120]]; // Bruno Berselli, Nov 27 2013

CROSSREFS

Cf. A001221 (primes counted without multiplicity), A046660, A144494, A074946, A134334. 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

Cf. A079149 (primes adj. to integers with at most 2 prime factors, A001222(n)<=2)

Sequence in context: A122810 A179953 A086436 * A098893 A069248 A008481

Adjacent sequences:  A001219 A001220 A001221 * A001223 A001224 A001225

KEYWORD

nonn,easy,nice,core

AUTHOR

N. J. A. Sloane, Apr 30 1991

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified August 28 19:05 EDT 2014. Contains 246177 sequences.