What Is A Distinct Prime Factor Ization, Unique Prime Factorization

The distinct prime components of a hopeful integer

=2" /> are defined as the

number

, ...,

in the prime factorization

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

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A perform of distinctive prime determinants of a number

deserve to be computed in the sdrta.net Language utilizing FactorInteger<>, and also the number

of unique prime components is implemented 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 also Stegun 1972, Kac 1959). This succession is offered by the train station Möbius transform of

, wherein

is the characteristic duty of the prime numbers (Sloane and Plouffe 1995, p.22). The element factorizations and also distinct prime components of the first couple of positive integers are provided in the table below.

	prime factorization		distinct prime components (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 only of distinctive prime factors are exactly the squarefreenumbers.

A sum involving

is offered by

(2)

for

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

The mean order that

(3)

(Hardy 1999, p.51). More precisely,

(4)

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

is the Mertens consistent and

space Stieltjes constants. Furthermore, the variance is given by

(5)

where

(6)
(7)

(OEIS A091588), where

(8)

(OEIS A085548) is the prime zeta role

(Finch 2003). The coefficients

and also

are given 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).

is a primorial, then

(18)

(Hardy and Wright 1979, p.355).

The summatory function of

is provided by

(19)

where

is the Mertens consistent (Hardy 1999, p.57), the

hatchet (Hardy and Ramanujan 1917; Hardy and Wright 1979, p.355) has been rewritten in a an ext explicit form, and also

and

space asymptotic notation. The first few values the the summatory duty 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 couple of numbers

which are products of one odd variety of distinct prime components (Hardy 1999, p.64; Ramanujan 2000, pp.xxiv and also 21) are 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 Stegun, I.A. (Eds.). Handbook of Mathematical functions with Formulas, Graphs, and also Mathematical Tables, ninth printing. New York: Dover, p.844, 1972.

Diaconis, P. "Asymptotic Expansions for the Mean and Variance that the number of Prime components 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 on Subjects suggested by His Life and also Work, third ed. Brand-new York: Chelsea, 1999.

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

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

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

" and also "The common Order that

and

." §22.10 and 22.11 in An advent to the concept of Numbers, 5th ed. Oxford, England: Clarendon Press, pp.354-358, 1979.

Kac, M. Statistical self-reliance in Probability, evaluation and Number Theory. Washington, DC: Math. Assoc. Amer., p.64, 1959.

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

Knuth, D.E. "Asymptotics because that

and

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

Ramanujan, S. Gathered Papers that 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 creature Sequences."

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

Referenced on sdrta.net|Alpha: distinct Prime Factors

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What Is A Distinct Prime Factor