A184388 -id:A184388 - 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

A206032 a(n) = Product_{d|n} sigma(d) where sigma = A000203.

Original entry on oeis.org

1, 3, 4, 21, 6, 144, 8, 315, 52, 324, 12, 28224, 14, 576, 576, 9765, 18, 73008, 20, 95256, 1024, 1296, 24, 25401600, 186, 1764, 2080, 225792, 30, 26873856, 32, 615195, 2304, 2916, 2304, 1302170688, 38, 3600, 3136, 128595600, 42, 84934656, 44, 762048, 584064

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Sequence is not the same as A206031(n): a(66) = 429981696, A206031(66) = 35831808.

In sequence a(n) are multiplied all values of sigma(d) of all divisors d of numbers n, in sequence A206031 are multiplied only distinct values of sigma(d) of all divisors d of numbers n.

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values k of sigma(d): 1,3,4,12; a(6) = Product of k = 1*3*4*12 = 144. For n=66 -> divisors d of 66: 1,2,3,6,11,22,33,66; corresponding values k of sigma(d): 1,3,4,12,12,36,48,144; a(66) = Product of k = 1*3*4*12*12*36*48*144 = 429981696.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A184388, A206031, A206033.

Table[Times @@ DivisorSigma[1, Divisors[n]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

a(n)=my(d=divisors(n));prod(i=2,#d,sigma(d[i])) \\ Charles R Greathouse IV, Feb 19 2013

a(p) = p+1, a(pq) = ((p+1)*(q+1))^2 for p, q = distinct primes.

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

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).

A184391 a(n) = product of numbers from 1 to A007955(n).

Original entry on oeis.org

1, 2, 6, 40320, 120, 371993326789901217467999448150835200000000, 5040, 126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000, 10888869450418352160768000000

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

Robert P. P. McKone, Table of n, a(n) for n = 1..11

Cf. A184387, A184388, A184389, A184390, A130674.

A184391[n_] := Product[i, {i, 1, n^(DivisorSigma[0, n]/2)}]; Table[A184391[n], {n, 1, 9}] (* Robert P. P. McKone, Feb 04 2021 *)

from math import isqrt, factorial
from sympy import divisor_count
def A184391(n): return factorial((isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2)) # Chai Wah Wu, Jun 25 2022

a(n) = A007955(n)! = A000142(A007955(n)).

a(9) from Robert P. P. McKone, Feb 04 2021

A206031 a(n) = product of numbers k <= sigma(n) such that k = sigma(d) for any divisor d of n where sigma = A000203.

Original entry on oeis.org

1, 3, 4, 21, 6, 144, 8, 315, 52, 324, 12, 28224, 14, 576, 576, 9765, 18, 73008, 20, 95256, 1024, 1296, 24, 25401600, 186, 1764, 2080, 225792, 30, 26873856, 32, 615195, 2304, 2916, 2304, 1302170688, 38, 3600, 3136, 128595600, 42, 84934656, 44, 762048, 584064

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

Sequence is not the same as A206032(n): a(66) = 35831808, A206032(66) = 429981696.

In sequence A206032 are multiplied all values of sigma(d) of all divisors d of numbers n, in sequence a(n) are multiplied only distinct values of sigma(d) of all divisors d of numbers n.

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values of sigma(d): 1,3,4,12; a(6) = Product of k = 1*3*4*12 = 144. For n=66 -> divisors d of 66: 1,2,3,6,11,22,33,66; corresponding values of sigma(d): 1,3,4,12,12,36,48,144; a(66) = Product of k = 1*3*4*12*36*48*144 = 35831808.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A184388, A184395, A206032, A206033.

Table[Times @@ Union[DivisorSigma[1, Divisors[n]]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

a(n)=my(d=vecsort(apply(sigma,divisors(n)),,8));prod(i=2,#d,d[i]) \\ Charles R Greathouse IV, Feb 19 2013

a(p) = p+1, a(pq) = ((p+1)*(q+1))^2 for p, q = distinct primes.

A206033 a(1) =1; for n>=1: a(n) = product of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n where sigma = A000203.

Original entry on oeis.org

1, 2, 6, 240, 120, 3326400, 5040, 4151347200, 119750400, 19760412672000, 39916800, 10802449851605508096000000, 6227020800, 1077167364120207360000, 1077167364120207360000, 842072570832352567099392000000, 355687428096000

Offset: 1

Jaroslav Krizek, Feb 03 2012

nonn

In sequence A206032 are multiplied all values of sigma(d) of all divisors d of numbers n, in sequence A206031 are multiplied only distinct values of sigma(d) of all divisors d of numbers n and in sequence a(n) are multiplied numbers k (1<=k<=sigma(n)) such that sigma(d) = k has no solution for neither divisor d of number n.

			For n=6 -> divisors d of 6: 1,2,3,6; corresponding values of sigma(d): 1,3,4,12; a(6) = Product of k = 2*5*6*7*8*9*10*11 = 3326400.

Cf. A184388, A206031, A206032.

Table[Times @@ Complement[Range[DivisorSigma[1, n]], DivisorSigma[1, Divisors[n]]], {n, 100}] (* T. D. Noe, Feb 10 2012 *)

A309548 Numbers k such that sigma(k)! - 1 is prime, where sigma is A000203.

Original entry on oeis.org

2, 3, 4, 5, 6, 11, 13, 21, 29, 31, 37, 170, 180, 214, 234, 265, 362, 369, 10734, 14318, 19679, 19876, 39636, 48784, 62517, 76225, 77277, 83629, 85519, 90649, 92287

Offset: 1

Hauke Löffler, Aug 07 2019

			2 is a term because sigma(2) = 3. 3! - 1 = 5, a prime.
6 is a term because sigma(6) = 12. 12! - 1 = 479001599, a prime.

Cf. A000040, A000203, A002982, A055490, A184388.

isok(n) = isprime(sigma(n)!-1); \\ Michel Marcus, Aug 07 2019

[n for n in range(1,150) if is_prime(factorial(sigma(n))-1)]

a(12)-a(24) from Daniel Suteu, Aug 07 2019

a(25)-a(31) from Amiram Eldar, May 14 2023

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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A206032 a(n) = Product_{d|n} sigma(d) where sigma = A000203.

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

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

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A184391 a(n) = product of numbers from 1 to A007955(n).

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A206031 a(n) = product of numbers k <= sigma(n) such that k = sigma(d) for any divisor d of n where sigma = A000203.

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A206033 a(1) =1; for n>=1: a(n) = product of numbers k <= sigma(n) such that k is not equal to sigma(d) for any divisor d of n where sigma = A000203.

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