login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A181930 Table T(d,n), where T(d,n)/LCM(1..d) gives the probability that d is the n-th divisor of an integer. 1
1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 4, 4, 3, 1, 0, 0, 0, 4, 5, 1, 0, 16, 20, 12, 6, 5, 1, 0, 0, 0, 48, 20, 26, 10, 1, 0, 0, 96, 40, 52, 44, 36, 11, 1, 0, 0, 0, 72, 48, 66, 34, 22, 9, 1, 0, 576, 720, 392, 384, 188, 154, 70, 26, 9, 1, 0, 0, 0, 0, 0, 480, 848, 560 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

By probability is meant limit density on [1,n] as n grows without bound.

Equivalently: T(n,d) is LCM(1..d) times the asymptotic density of the set of natural numbers whose n-th divisor is equal to d.

Rows are (1), (0,1), (0,1,1), (0,0,2,1),....

LINKS

David W. Wilson, Table of n, a(n) for n = 1..820

FORMULA

T(d,d) = 1.

T(d,n) = 0 if n < tau(d) = A000005(d). If d is a divisor of n, then every divisor of d is also a divisor of n, and d is therefore at least the tau(n)-th divisor of n.

T(d,n) > 0 for all tau(d) <= n <= d. To show this, let S be the set of the divisors of d along with the smallest n-tau(d) non-divisors of d. Then S has n elements, the largest being d. The elements of S are the smallest divisors of LCM(S), so d is the n-th divisor of LCM(S). Hence every number of the form LCM(S) + k LCM(1..d) has n-th divisor d, so numbers with n-th divisor d have asymptotic density >= 1/LCM(1..d), and T(d,n) > 0.

SUM(d=1..inf; T(d,n)/LCM(1..d)) = 1.

SUM(n=1..d; T(d,n)/LCM(1..d)) = 1/d.

T(d,tau(d)) = LCM(1..d)/d * product( (q-1)/q, q prime and there is an a with q^a<d and q^a does not divide d). In particular, if p is prime, then T(p,2) = LCM(1..p)/p * product( (q-1)/q, q prime and q<d). - Benoit Jubin, Apr 02 2012

EXAMPLE

T(5,4) = 4th element of the fifth row (0,4,4,3,1) = 3. T(5,4)/LCM(1..5) = T(5,4)/A003418(5) = T(5,4)/60 = 1/20 is the probability that 5 is the 4th divisor of an integer.

CROSSREFS

Sequence in context: A049243 A077908 A052922 * A109167 A066426 A100887

Adjacent sequences:  A181927 A181928 A181929 * A181931 A181932 A181933

KEYWORD

nonn,tabl

AUTHOR

David W. Wilson, Apr 02 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 19 18:23 EST 2014. Contains 252239 sequences.