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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A208133 Total number of subgroups of rank <= 2 of a certain class of abelian groups of order n defined as direct products of Z/(mZ) X Z/(kZ) with m|k. 2
1, 2, 2, 8, 2, 4, 2, 12, 9, 4, 2, 16, 2, 4, 4, 31, 2, 18, 2, 16, 4, 4, 2, 24, 11, 4, 14, 16, 2, 8, 2, 42, 4, 4, 4, 72, 2, 4, 4, 24, 2, 8, 2, 16, 18, 4, 2, 62, 13, 22, 4, 16, 2, 28, 4, 24, 4, 4, 2, 32, 2, 4, 18, 90, 4, 8, 2, 16, 4, 8, 2, 108, 2, 4, 22, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Level function l_tau^2(n) of Bhowmik and Wu.

Records occur at 1, 2, 4, 8, 12, 16, 32, 36, 64, 72, 108, 128, 144, 288, 432, 576, 1152, 1296, 2304, 3600, 5184, 7200, 9216, 10368, 14112, 14400, 20736, 28224, 28800, 32400, 57600, ... and they are: 1, 2, 8, 12, 16, 31, 42, 72, 90, 108, 112, 116, 279, 378, 434, 810, 1044, 1302, 2025, 3069, 3780, 4158, 4644, 4872, 4914, 8910, 9450, 10530, 11484, 14322, 22275, ... - Antti Karttunen, Mar 21 2018

REFERENCES

A. Laurincikas, The universality of Dirichlet series attached to finite Abelian groups, in "Number Theory", Proc. Turku Sympos. on Number Theory, May 31-June 4, 1999, p 179.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

G. Bhowmik, Jie Wu, Zeta function of subgroups of abelian groups and average orders, J. reine angew. Math. 530 (2001) 1-15.

FORMULA

Dirichlet g.f.: zeta(s)^2*zeta(2s)^2*zeta(2s-1)*Product_{primes p} (1 + 1/p^(2s) - 2/p^(3s)).

MAPLE

L300828 := [ 1, 0, 0, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

] ;

L010052 := [ 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ];

L037213 := [ 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ;

Lx := DIRICHLET(L300828, L037213) ;

Lx := DIRICHLET(Lx, L010052) ;

Lx := DIRICHLET(Lx, L010052) ;

Lx := MOBIUSi(Lx) ;

Lx := MOBIUSi(Lx) ;

# Name of initial list L1 changed to L300828 to refer to sequence A300828 by Antti Karttunen, Mar 21 2018

PROG

(PARI)

A037213(n) = if(issquare(n), sqrtint(n), 0);

A300828(n) = { if(1==n, return(1)); my(val=1, v=factor(n), d=matsize(v)[1]); for(i=1, d, if(v[i, 2] < 2 || v[i, 2] > 3, return(0)); if (v[i, 2] == 3, val *= -2)); return(val); };

a208133s1(n) = sumdiv(n, d, A037213(n/d)*A300828(d));

a208133s2(n) = sumdiv(n, d, issquare(n/d)*a208133s1(d));

a208133s3(n) = sumdiv(n, d, issquare(n/d)*a208133s2(d));

a208133s4(n) = sumdiv(n, d, a208133s3(d));

A208133(n) = sumdiv(n, d, a208133s4(d)); \\ Antti Karttunen, Mar 21 2018, after R. J. Mathar's Maple code

CROSSREFS

Cf. A010052, A037213, A300828.

Sequence in context: A037300 A029623 A325753 * A046644 A161915 A174354

Adjacent sequences:  A208130 A208131 A208132 * A208134 A208135 A208136

KEYWORD

nonn,mult

AUTHOR

R. J. Mathar, Mar 29 2012

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 17 08:31 EDT 2019. Contains 327127 sequences. (Running on oeis4.)