The distinctive prime factors of a positive integer

=2" /> are characterized as the
numbers
, ...,
in the prime factorization

(Hardy and also Wright 1979, p.354).

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A list of distinctive prime components of a number

have the right to be computed in the dearteassociazione.org Language making use of FactorInteger<>, and also the number
of distinct prime factors is applied as PrimeNu.

The first couple of values the

because that
, 2, ... Space 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, ... (OEIS A001221; Abramowitz and Stegun 1972, Kac 1959). This succession is given by the station Möbius transform of
, wherein
is the characteristic duty of the prime numbers (Sloane and also Plouffe 1995, p.22). The prime factorizations and also distinct prime components of the first few positive integers are listed in the table below.

 prime factorization distinct prime factors (A027748) 1 -- 0 -- 2 2 1 2 3 3 1 3 4 1 2 5 5 1 5 6 2 2, 3 7 7 1 7 8 1 2 9 1 3 10 2 2, 5 11 11 1 11 12 2 2, 3 13 13 1 13 14 2 2, 7 15 2 3, 5 16 1 2

The numbers consisting just of distinct prime factors are specifically the squarefreenumbers.

A sum including

is given by

(2)

for

1" /> (Hardy and Wright 1979, p.255).

The mean order that

is

(3)

(Hardy 1999, p.51). An ext precisely,

(4)

(Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003, Knuth 2003), where

is the Mertens continuous and
space Stieltjes constants. Furthermore, the variance is given by

(5)

where

 (6)(7)

(OEIS A091588), where

(8)

(OEIS A085548) is the prime zeta function

(Finch 2003). The coefficients
and
are offered by the sums

 (9)(10)(11)(12)(13)

(Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003, Knuth 2003), where

 (14)(15)(16)(17)

(Finch 2003).

If

is a primorial, then

(18)

(Hardy and also Wright 1979, p.355).

The summatory role of

is offered by

(19)

where

is the Mertens consistent (Hardy 1999, p.57), the
ax (Hardy and also Ramanujan 1917; Hardy and Wright 1979, p.355) has been rewritten in a much more explicit form, and also
and
space asymptotic notation. The first few values that the summatory role are 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, ... (OEIS A013939). In addition,

(20)

(Hardy and also Wright 1979, p.357).

The first few numbers

which are products of an odd number of distinct prime determinants (Hardy 1999, p.64; Ramanujan 2000, pp.xxiv and also 21) space 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, ... (OEIS A030059).
satisfies

(21)

(Hardy 1999, pp.64-65). In addition, if

is the variety of
with
REFERENCES:

Abramowitz, M. And also Stegun, I.A. (Eds.). Handbook the Mathematical attributes with Formulas, Graphs, and also Mathematical Tables, 9th printing. New York: Dover, p.844, 1972.

Diaconis, P. "Asymptotic Expansions because that the Mean and Variance that the number of Prime determinants of a Number

." Dept. Statistics Tech. Report 96, Stanford, CA: Stanford University, 1976.

Diaconis, P. "G.H.Hardy and also Probability???" Bull. LondonMath. Soc. 34, 385-402, 2002.

Finch, S. "Two Asymptotic Series." December 10, 2003. Http://algo.inria.fr/bsolve/.

Hardy, G.H. Ramanujan: Twelve Lectures ~ above Subjects suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.

Hardy, G.H. And Ramanujan, S. "The Normal variety of Prime determinants of a Number

." Quart. J. Math. 48, 76-92, 1917.

Hardy, G.H. And Wright, E.M. "The number of Prime factors of

" and "The regular Order the
and
." §22.10 and 22.11 in An arrival to the theory of Numbers, fifth ed. Oxford, England: Clarendon Press, pp.354-358, 1979.

Kac, M. Statistical freedom in Probability, analysis and Number Theory. Washington, DC: Math. Assoc. Amer., p.64, 1959.

Knuth, D.E. Selected papers on evaluation of Algorithms. Stanford, CA: CSLI Publications, pp.338-339, 2000.

Knuth, D.E. "Asymptotics for

and
." quote by Finch (2003). Unpublished note, 2003.

Ramanujan, S. Accumulated Papers of Srinivasa Ramanujan (Ed. G.H.Hardy, P.V.S.Aiyar, and also B.M.Wilson). Providence, RI: Amer. Math. Soc., 2000.

Sloane, N.J.A. Order A001221/M0056, A013939, A027748, A085548, and also A091588 in "The On-Line Encyclopedia of integer Sequences."

Sloane, N.J.A. And Plouffe, S. TheEncyclopedia of creature Sequences. San Diego, CA: academic Press, 1995.

Referenced on dearteassociazione.org|Alpha: distinctive Prime Factors

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