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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A055155 a(n) = Sum_{k|n} gcd(k, n/k). 5
1, 2, 2, 4, 2, 4, 2, 6, 5, 4, 2, 8, 2, 4, 4, 10, 2, 10, 2, 8, 4, 4, 2, 12, 7, 4, 8, 8, 2, 8, 2, 14, 4, 4, 4, 20, 2, 4, 4, 12, 2, 8, 2, 8, 10, 4, 2, 20, 9, 14, 4, 8, 2, 16, 4, 12, 4, 4, 2, 16, 2, 4, 10, 22, 4, 8, 2, 8, 4, 8, 2, 30, 2, 4, 14, 8, 4, 8, 2, 20, 17, 4, 2, 16, 4, 4, 4, 12, 2, 20, 4, 8, 4, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is odd iff n is odd square. - Vladeta Jovovic, Aug 27 2002

From Robert Israel, Dec 26 2015: (Start)

a(n) >= A000005(n), with equality iff n is squarefree (i.e., is in A005117).

a(n) = 2 iff n is prime. (End)

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

E. Krätzel, W. G. Nowak, L. Tóth, On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., 10 (2012), 761-774.

M. Kühleitner, W. G. Nowak, On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions , arXiv: 1204.1146 [math.NT] (2012).

Project Euler, Problem 530 - GCD of Divisors.

L. Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.

FORMULA

Multiplicative with a(p^e) = (p^(e/2)*(p+1)-2)/(p-1) for even e and a(p^e) = 2*(p^((e+1)/2)-1)/(p-1) for odd e. - Vladeta Jovovic, Nov 01 2001

Dirichlet g.f. (zeta(s))^2*zeta(2s-1)/zeta(2s); inverse Mobius transform of A000188. - R. J. Mathar, Feb 16 2011

Dirichlet convolution of A069290 and A008966. - R. J. Mathar, Oct 31 2011

Sum_{k=1..n} a(k) ~ 3*n / (2*Pi^6) * (Pi^4 * log(n)^2 + ((8*g - 2)*Pi^4 - 24 * Pi^2 * z1) * log(n) + 2*Pi^4 * (1 - 4*g + 5*g^2 - 6*sg1) + 288 * z1^2 - 24 * Pi^2 * (-z1 + 4*g*z1 + z2)), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant A082633, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994. - Vaclav Kotesovec, Feb 01 2019

EXAMPLE

a(9) = gcd(1,9) + gcd(3,3) + gcd(9,1) = 5, since 1, 3, 9 are the positive divisors of 9.

MAPLE

N:= 1000: # to get a(1) to a(N)

V:= Vector(N):

for k from 1 to N do

   for j from 1 to floor(N/k) do

     V[k*j]:= V[k*j]+igcd(k, j)

   od

od:

convert(V, list); # Robert Israel, Dec 26 2015

MATHEMATICA

Table[DivisorSum[n, GCD[#, n/#] &], {n, 94}] (* Michael De Vlieger, Sep 23 2017 *)

PROG

(PARI) a(n) = sumdiv(n, d, gcd(d, n/d)); \\ Michel Marcus, Aug 03 2016

CROSSREFS

Cf. A000005, A005117, A057670.

Sequence in context: A322327 A124315 A101113 * A085191 A188581 A318316

Adjacent sequences:  A055152 A055153 A055154 * A055156 A055157 A055158

KEYWORD

easy,nonn,mult

AUTHOR

Leroy Quet, Jul 02 2000

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 April 18 14:46 EDT 2019. Contains 322209 sequences. (Running on oeis4.)