A184389 a(n) = Sum_{k=1..tau(n)} k, where tau is the number of divisors of n (A000005).

1, 3, 3, 6, 3, 10, 3, 10, 6, 10, 3, 21, 3, 10, 10, 15, 3, 21, 3, 21, 10, 10, 3, 36, 6, 10, 10, 21, 3, 36, 3, 21, 10, 10, 10, 45, 3, 10, 10, 36, 3, 36, 3, 21, 21, 10, 3, 55, 6, 21, 10, 21, 3, 36, 10, 36, 10, 10, 3, 78, 3, 10, 21, 28, 10, 36, 3, 21, 10, 36, 3, 78

Offset

1,2

Comments

Length of row n in triangle A187207. - Omar E. Pol, Aug 07 2011

Number of pairs of even divisors of 2n, (d1,d2), such that d1<=d2. - Wesley Ivan Hurt, Aug 24 2020

Links

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Formula

a(n) = A000217(A000005(n)) = (1/2)*A000005(n)*(A000005(n)+1).

a(n) = A066446(n) + A000005(n) = A035116(n) - A066446(n). - Reinhard Zumkeller, Sep 08 2015

Dirichlet g.f.: zeta(s)^2*(zeta(s)^2 + zeta(2*s))/(2*zeta(2*s)). - Ilya Gutkovskiy, Jun 25 2016

a(n) = Sum_{d1|(2*n), d2|(2*n), d1 and d2 even, d1<=d2} 1. - Wesley Ivan Hurt, Aug 24 2020

a(n) = Sum_{d|n} A018892(d). - Daniel Suteu, Jan 08 2021

a(n) = Sum_{d|n} A135539(n,d). - Ridouane Oudra, May 29 2025

a(n) = A337362(n) + A129308(n). - Ridouane Oudra, May 30 2025

Example

For n = 4; tau(4) = 3; a(4) = 1+2+3 = 6.

Maple

A184389 := proc(n)
    A000217(numtheory[tau](n)) ;
end proc: # R. J. Mathar, Oct 04 2014

Mathematica

((#+1)#)/2&/@DivisorSigma[0, Range[80]] (* Harvey P. Dale, Feb 27 2013 *)

Prog

(Haskell)
a184389 = a000217 . a000005'  -- Reinhard Zumkeller, Sep 08 2015
(PARI) a(n) = my(nd=numdiv(n)); nd*(nd+1)/2; \\ Michel Marcus, Jun 25 2016
(Python)
from sympy import divisor_count
def A184389(n): return (d:=divisor_count(n))*(d+1)>>1 # Chai Wah Wu, Jan 29 2026

Crossrefs

Cf. A000005 (tau), A000217 (triangular numbers).

Cf. A184387, A184388, A184390, A184391, A130674.

Cf. A035116, A066446.

Cf. A187207, A135539, A337362, A129308.

Sequence in context: A112163 A058587 A196439 * A163163 A376646 A378820

Adjacent sequences: A184386 A184387 A184388 * A184390 A184391 A184392

Keyword

nonn

Author

Jaroslav Krizek, Jan 12 2011

Status

approved