A184391 -id:A184391 - cp's OEIS frontend

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

Jaroslav Krizek, Jan 12 2011

nonn

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

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

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

Cf. A000005 (tau), A000217 (triangular numbers).
Cf. A184387, A184388, A184390, A184391, A130674.
Cf. A035116, A066446.
Cf. A187207, A135539, A337362, A129308.

a184389 = a000217 . a000005'  -- Reinhard Zumkeller, Sep 08 2015

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

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

a(n) = my(nd=numdiv(n)); nd*(nd+1)/2; \\ Michel Marcus, Jun 25 2016

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

A184387 a(n) = sum of numbers from 1 to sigma(n), where sigma(n) = A000203(n).

Original entry on oeis.org

1, 6, 10, 28, 21, 78, 36, 120, 91, 171, 78, 406, 105, 300, 300, 496, 171, 780, 210, 903, 528, 666, 300, 1830, 496, 903, 820, 1596, 465, 2628, 528, 2016, 1176, 1485, 1176, 4186, 741, 1830, 1596, 4095, 903, 4656, 990, 3570, 3081, 2628, 1176, 7750, 1653, 4371

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

			For n = 4; sigma(4) = 7; a(4) = 1+2+3+4+5+6+7 = 28.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A000217, A002117, A184388, A184389, A184390, A184391, A130674.

Array[PolygonalNumber@ DivisorSigma[1, #] &, 50] (* Michael De Vlieger, Nov 16 2017 *)

a(n)=binomial(sigma(n)+1,2) \\ Charles R Greathouse IV, Feb 14 2013

from sympy import divisor_sigma
def A184387(n): return (m:=divisor_sigma(n))*(m+1)>>1 # Chai Wah Wu, Jul 05 2023

a(n) = Sum_{i = 1..sigma(n)} i = A000217(A000203(n)) = A000203(n)*(A000203(n) + 1)/2.

Sum_{k=1..n} a(k) = (5*zeta(3)/12) * n^3 + O(n^2*log(n)^2). - Amiram Eldar, Dec 08 2022

A184388 a(n) = product of numbers from 1 to sigma(n), where sigma(n) = A000203(n).

Original entry on oeis.org

1, 6, 24, 5040, 720, 479001600, 40320, 1307674368000, 6227020800, 6402373705728000, 479001600, 304888344611713860501504000000, 87178291200, 620448401733239439360000, 620448401733239439360000, 8222838654177922817725562880000000, 6402373705728000

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

			For n = 4; sigma(4) = 7; a(n) = 1*2*3*4*5*6*7 = 5040.

Cf. A184387, A184389, A184390, A184391, A130674.

DivisorSigma[1,Range[20]]! (* Harvey P. Dale, Apr 09 2019 *)

a(n)=sigma(n)! \\ Charles R Greathouse IV, Feb 13 2013

a(n) = sigma(n)! = Product_(i = 1,…,sigma(n)) i = A000142(A000203(n)) = (A000203(n))!.

Corrected and extended by Harvey P. Dale, Apr 09 2019

A184390 a(n) = sum of numbers from 1 to pi(n), where pi(n) = A007955(n).

Original entry on oeis.org

1, 3, 6, 36, 15, 666, 28, 2080, 378, 5050, 66, 1493856, 91, 19306, 25425, 524800, 153, 17009028, 190, 32004000, 97461, 117370, 276, 55037822976, 7875, 228826, 266085, 240956128, 435, 328050405000

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

			For n = 6; pi(6) = 36; a(n) = (1/2)*36*37 = 666.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A184387, A184388, A184389, A184391, A130674.

# (#+1)/2&/@Array[Times@@Divisors[#]&,40] (* Harvey P. Dale, Oct 05 2012 *)

from math import isqrt
from sympy import divisor_count
def A184390(n): return (m:=((isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2)))*(m+1)//2 # Chai Wah Wu, Jun 25 2022

a(n) = Sum_{i = 1..pi(n)} i = A000217(A007955(n)) = (1/2)*A007955(n)*(A007955(n)+1).

cp's OEIS Frontend

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

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A184387 a(n) = sum of numbers from 1 to sigma(n), where sigma(n) = A000203(n).

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A184388 a(n) = product of numbers from 1 to sigma(n), where sigma(n) = A000203(n).

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Mathematica

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A184390 a(n) = sum of numbers from 1 to pi(n), where pi(n) = A007955(n).

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