A283078 -id:A283078 - cp's OEIS frontend

A062731 Sum of divisors of 2*n.

3, 7, 12, 15, 18, 28, 24, 31, 39, 42, 36, 60, 42, 56, 72, 63, 54, 91, 60, 90, 96, 84, 72, 124, 93, 98, 120, 120, 90, 168, 96, 127, 144, 126, 144, 195, 114, 140, 168, 186, 126, 224, 132, 180, 234, 168, 144, 252, 171, 217, 216, 210, 162, 280, 216, 248, 240, 210

Offset: 1

Jason Earls, Jul 11 2001

a(n) is also the total number of parts in all partitions of 2*n into equal parts. - Omar E. Pol, Feb 14 2021

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from Harry J. Smith]

Sigma(k*n): A000203 (k=1), A144613 (k=3), A193553 (k=4, even bisection), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).
Cf. A008438, A074400, A182818, A239052 (odd bisection), A326124 (partial sums), A054784, A215947, A336923, A346870, A346878, A346880, A355750.
Row 2 of A319526. Column & Row 2 of A216626. Row 1 of A355927.
Shallow diagonal (2n,n) of A265652. See also A244658.

[SumOfDivisors(2*n): n in [1..70]]; // Vincenzo Librandi, Oct 31 2014

lst={};Do[AppendTo[lst, DivisorSigma[1, n]], {n, 2, 6!, 2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 20 2008 *)
DivisorSigma[1,2*Range[60]] (* Harvey P. Dale, Jun 08 2022 *)

numlib::sigma(2*n)$ n=0..81 // Zerinvary Lajos, May 13 2008

PARI
```
vector(66,n,sigma(2*n,1))
```

for (n=1, 1000, write("b062731.txt", n, " ", sigma(2*n)) ) \\ Harry J. Smith, Aug 09 2009

a(n) = A000203(2*n). - R. J. Mathar, Apr 06 2011
a(n) = A000203(n) + A054785(n). - R. J. Mathar, May 19 2020
From Vaclav Kotesovec, Aug 07 2022: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-1) * (3 - 2^(1-s)).
Sum_{k=1..n} a(k) ~ 5 * Pi^2 * n^2 / 24. (End)
From Miles Wilson, Sep 30 2024: (Start)
G.f.: Sum_{k>=1} k*x^(k/gcd(k, 2))/(1 - x^(k/gcd(k, 2))).
G.f.: Sum_{k>=1} k*x^(2*k/(3 + (-1)^k))/(1 - x^(2*k/(3 + (-1)^k))). (End)

Zero removed and offset corrected by Omar E. Pol, Jul 17 2009

A224613 a(n) = sigma(6*n).

Original entry on oeis.org

12, 28, 39, 60, 72, 91, 96, 124, 120, 168, 144, 195, 168, 224, 234, 252, 216, 280, 240, 360, 312, 336, 288, 403, 372, 392, 363, 480, 360, 546, 384, 508, 468, 504, 576, 600, 456, 560, 546, 744, 504, 728, 528, 720, 720, 672, 576, 819, 684, 868, 702, 840, 648

Offset: 1

Zak Seidov, Apr 22 2013

nonn

Conjectures: sigma(6n) > sigma(6n - 1) and sigma(6n) > sigma(6n + 1).
Conjectures are false. Try prime 73961483429 for n. One finds sigma(6*73961483429) < sigma(6*73961483429+1). The number n = 105851369791 provides a counterexample for the other case. - T. D. Noe, Apr 22 2013
Sum of the divisors of the numbers k which have the property that the width associated to the vertex of the first (also the last) valley of the smallest Dyck path of the symmetric representation of sigma(k) is equal to 2 (see example). Other positive integers have width 0 or 1 associated to the mentioned valley. - Omar E. Pol, Aug 11 2021

			From _Omar E. Pol_, Aug 11 2021: (Start)
Illustration of initial terms:
----------------------------------------------------------------------
   n    6*n   a(n)    Diagram:  1           2           3           4
----------------------------------------------------------------------
                                _           _           _           _
                               | |         | |         | |         | |
                               | |         | |         | |         | |
                          * _ _| |         | |         | |         | |
                           |  _ _|         | |         | |         | |
                      _ _ _| |_|           | |         | |         | |
   1     6     12    |_ _ _ _|      * _ _ _| |         | |         | |
                                    _|  _ _ _|         | |         | |
                                * _|  _| |             | |         | |
                                 |  _|  _|    * _ _ _ _| |         | |
                                 | |_ _|       |  _ _ _ _|         | |
                      _ _ _ _ _ _| |          _| | |               | |
   2    12     28    |_ _ _ _ _ _ _|        _|  _|_|    * _ _ _ _ _| |
                                      * _ _|  _|         |  _ _ _ _ _|
                                       |  _ _|        _ _| | |
                                       | |_ _|      _|  _ _| |
                                       | |        _|  _|  _ _|
                      _ _ _ _ _ _ _ _ _| |       |  _|  _|
   3    18     39    |_ _ _ _ _ _ _ _ _ _|  * _ _| |  _|
                                             |  _ _| |
                                             | |_ _ _|
                                             | |
                                             | |
                      _ _ _ _ _ _ _ _ _ _ _ _| |
   4    24     60    |_ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the mentioned vertices are aligned on two straight lines that meet at point (3,3).
a(n) equals the area (also the number of cells) in the n-th diagram. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), A283118 (k=5), this sequence (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).
Cf. A000203 (sigma(n)), A053224 (n: sigma(n) < sigma(n+1)).
Cf. A067825 (even n: sigma(n)< sigma(n+1)).
Cf. A008588, A237593.

DivisorSigma[1,6*Range[60]] (* Harvey P. Dale, Apr 16 2016 *)

a(n)=sigma(6*n) \\ Charles R Greathouse IV, Apr 22 2013

from sympy import divisor_sigma
def a(n):  return divisor_sigma(6*n)
print([a(n) for n in range(1, 54)]) # Michael S. Branicky, Dec 28 2021

from math import prod
from collections import Counter
from sympy import factorint
def A224613(n): return prod((p**(e+1)-1)//(p-1) for p, e in (Counter(factorint(n))+Counter([2,3])).items()) # Chai Wah Wu, Sep 07 2023

a(n) = A000203(6n).
a(n) = A000203(A008588(n)). - Omar E. Pol, Aug 11 2021
Sum_{k=1..n} a(k) = (55*Pi^2/72) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

Corrected by Harvey P. Dale, Apr 16 2016

A193553 Sum of divisors of 4*n.

Original entry on oeis.org

7, 15, 28, 31, 42, 60, 56, 63, 91, 90, 84, 124, 98, 120, 168, 127, 126, 195, 140, 186, 224, 180, 168, 252, 217, 210, 280, 248, 210, 360, 224, 255, 336, 270, 336, 403, 266, 300, 392, 378, 294, 480, 308, 372, 546, 360, 336, 508, 399, 465, 504, 434, 378, 600, 504, 504, 560, 450, 420, 744, 434, 480, 728, 511, 588, 720

Offset: 1

Joerg Arndt, Jul 30 2011

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), this sequence (k=4), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).

Cf. A008438, A008586, A182820.

DivisorSigma[1,4*Range[70]] (* Harvey P. Dale, Jan 27 2015 *)

PARI
```
vector(66, n, sigma(4*n, 1))
```

a(n) = sigma(4*n) = A000203(4*n).
a(n) = 3*sigma(2*n) - 2*sigma(n); the relation is the special case e=1, p=2 of the relation sigma(t^2*n) = (t+1)*sigma(t*n) - t*sigma(n) where t=p^e (p a prime).
G.f. is x times the logarithmic derivative of the g.f. of A182820.
a(2*n-1) = 7 * A008438(n) = 7 * sigma(2*n-1); special case of sigma(2^k*(2*n-1)) = (2^(k+1)-1) * sigma(2*n-1).
Sum_{k=1..n} a(k) = (11*Pi^2/24) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022
G.f.: Sum_{k>=1} k*x^(k/gcd(k, 4))/(1 - x^(k/gcd(k, 4))). - Miles Wilson, Sep 29 2024

A144613 a(n) = sigma(3n) = A000203(3n).

Original entry on oeis.org

4, 12, 13, 28, 24, 39, 32, 60, 40, 72, 48, 91, 56, 96, 78, 124, 72, 120, 80, 168, 104, 144, 96, 195, 124, 168, 121, 224, 120, 234, 128, 252, 156, 216, 192, 280, 152, 240, 182, 360, 168, 312, 176, 336, 240, 288, 192, 403, 228, 372, 234, 392, 216, 363, 288, 480, 260, 360

Offset: 1

N. J. A. Sloane, Jan 15 2009

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), this sequence (k=3), A193553 (k=4), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).

Cf. A008585, A078708.

a[n_] := DivisorSigma[1, 3*n]; Array[a, 60] (* Amiram Eldar, Dec 16 2022 *)

vector(66, n, sigma(3*n, 1)) \\ Joerg Arndt, Jul 30 2011

a(n) = A000203(n) + 3*A078708(n). - R. J. Mathar, May 19 2020

Sum_{k=1..n} a(k) = (11*Pi^2/36) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

Zero removed and offset corrected by Seiichi Manyama, Feb 28 2017

A283122 a(n) = sigma(8*n).

Original entry on oeis.org

15, 31, 60, 63, 90, 124, 120, 127, 195, 186, 180, 252, 210, 248, 360, 255, 270, 403, 300, 378, 480, 372, 360, 508, 465, 434, 600, 504, 450, 744, 480, 511, 720, 558, 720, 819, 570, 620, 840, 762, 630, 992, 660, 756, 1170, 744, 720, 1020, 855, 961

Offset: 1

Seiichi Manyama, Mar 01 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), A283118 (k=5), A224613 (k=6), A283078 (k=7), this sequence (k=8), A283123 (k=9).

Cf. A008590.

Table[DivisorSigma[1, 8 n], {n, 50}] (* Michael De Vlieger, Mar 01 2017 *)

a(n) = sigma(8*n) \\ Amiram Eldar, Dec 16 2022

a(n) = A000203(8*n).

Sum_{k=1..n} a(k) = (23*Pi^2/24) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A283123 a(n) = sigma(9*n).

Original entry on oeis.org

13, 39, 40, 91, 78, 120, 104, 195, 121, 234, 156, 280, 182, 312, 240, 403, 234, 363, 260, 546, 320, 468, 312, 600, 403, 546, 364, 728, 390, 720, 416, 819, 480, 702, 624, 847, 494, 780, 560, 1170, 546, 960, 572, 1092, 726, 936, 624, 1240, 741, 1209

Offset: 1

Seiichi Manyama, Mar 01 2017

nonn

In general, for k>=1, Sum_{j=1..n} sigma(j*k) ~ A069097(k) * Pi^2 * n^2 / (12*k). - Vaclav Kotesovec, May 11 2024

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), this sequence (k=9).

Cf. A008591.

Table[DivisorSigma[1, 9 n], {n, 50}] (* Michael De Vlieger, Mar 01 2017 *)

a(n) = sigma(9*n) \\ Amiram Eldar, Dec 16 2022

a(n) = A000203(9*n).

Sum_{k=1..n} a(k) = (35*Pi^2/36) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A283118 a(n) = sigma(5*n).

Original entry on oeis.org

6, 18, 24, 42, 31, 72, 48, 90, 78, 93, 72, 168, 84, 144, 124, 186, 108, 234, 120, 217, 192, 216, 144, 360, 156, 252, 240, 336, 180, 372, 192, 378, 288, 324, 248, 546, 228, 360, 336, 465, 252, 576, 264, 504, 403, 432, 288, 744, 342, 468, 432, 588, 324, 720

Offset: 1

Seiichi Manyama, Mar 01 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), this sequence (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).

Cf. A008587.

Table[DivisorSigma[1, 5 n], {n, 56}] (* Michael De Vlieger, Mar 01 2017 *)

a(n) = sigma(5*n) \\ Amiram Eldar, Dec 16 2022

a(n) = A000203(5*n).

Sum_{k=1..n} a(k) = (29*Pi^2/60) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A319528 a(n) = 8 * sigma(n).

Original entry on oeis.org

8, 24, 32, 56, 48, 96, 64, 120, 104, 144, 96, 224, 112, 192, 192, 248, 144, 312, 160, 336, 256, 288, 192, 480, 248, 336, 320, 448, 240, 576, 256, 504, 384, 432, 384, 728, 304, 480, 448, 720, 336, 768, 352, 672, 624, 576, 384, 992, 456, 744, 576, 784, 432, 960, 576, 960, 640, 720, 480, 1344, 496, 768, 832

Offset: 1

Omar E. Pol, Sep 22 2018

8 times the sum of the divisors of n.

a(n) is also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) in which the structure of every 45-degree three-dimensional sector arises after the 45-degree zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is an eight-pointed star formed by eight rhombuses (see Links section).

Muniru A Asiru, Table of n, a(n) for n = 1..10000
Omar E. Pol, Diagram of the triangle A237593 before the 45-degree-zig-zag folding (rows: 1..28)
Index entries for sequences related to sigma(n)

k times sigma(n), k=1..7: A000203, A074400, A272027, A239050, A274535, A274536, A319527.

Cf. A008589, A047304, A237593, A283078.

List([1..70],n->8*Sigma(n)); # Muniru A Asiru, Sep 28 2018

with(numtheory): seq(8*sigma(n), n=1..64);

8*DivisorSigma[1,Range[70]] (* Harvey P. Dale, Dec 24 2018 *)

PARI
```
a(n) = 8 * sigma(n);
```

a(n) = 8*A000203(n) = 4*A074400(n) = 2*A239050(n).
a(n) = A000203(n) + A319527(n).
Dirichlet g.f.: 8*zeta(s-1)*zeta(s). (After Ilya Gutkovskiy)
Conjecture: a(n) = sigma(7*n) = A283078(n) iff n is not a multiple of 7.
Conjecture is true, since sigma is multiplicative, so if (7,n) = 1 then sigma(7*n) = sigma(7)*sigma(n) = 8*sigma(n). - Charlie Neder, Oct 02 2018

A372787 a(n) = tau(7n) = A000005(7n).

Original entry on oeis.org

2, 4, 4, 6, 4, 8, 3, 8, 6, 8, 4, 12, 4, 6, 8, 10, 4, 12, 4, 12, 6, 8, 4, 16, 6, 8, 8, 9, 4, 16, 4, 12, 8, 8, 6, 18, 4, 8, 8, 16, 4, 12, 4, 12, 12, 8, 4, 20, 4, 12, 8, 12, 4, 16, 8, 12, 8, 8, 4, 24, 4, 8, 9, 14, 8, 16, 4, 12, 8, 12, 4, 24, 4, 8, 12, 12, 6, 16, 4

Offset: 1

Vaclav Kotesovec, May 13 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A283078.

Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372788, A372789, A372790, A372791, A372792.

Table[DivisorSigma[0, 7*n], {n, 1, 150}]

Sum_{k=1..n} a(k) ~ (13*n*(log(n) + 2*gamma - 1) + n*log(7)) / 7, where gamma is the Euler-Mascheroni constant A001620.

A088842 Denominator of the quotient sigma(7n)/sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 8

Offset: 1

Labos Elemer, Nov 04 2003

Sum of powers of 7 dividing n. - Amiram Eldar, Nov 27 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A001620, A088841 (numerators), A283078 (sigma(7n)).

Cf. A080278, A038712, A088837, A088838, A088839, A088840.

Table[Denominator[DivisorSigma[1, 7*n]/DivisorSigma[1, n]], {n, 1, 128}] (* corrected by Ilya Gutkovskiy, Dec 15 2020 *)
a[n_] := (7^(IntegerExponent[n, 7] + 1) - 1)/6; Array[a, 100] (* Amiram Eldar, Nov 27 2022 *)

a(n) = denominator(sigma(7*n)/sigma(n)); \\ Michel Marcus, Dec 15 2020

a(n) = (7^(valuation(n, 7) + 1) - 1)/6; \\ Amiram Eldar, Nov 27 2022

G.f.: Sum_{k>=0} 7^k * x^(7^k) / (1 - x^(7^k)). - Ilya Gutkovskiy, Dec 15 2020
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(7^e) = (7^(e+1)-1)/6, and a(p^e) = 1 for p != 7.
Dirichlet g.f.: zeta(s) / (1 - 7^(1 - s)).
Sum_{k=1..n} a(k) ~ n*log_7(n) + (1/2 + (gamma - 1)/log(7))*n, where gamma is Euler's constant (A001620). (End)

cp's OEIS Frontend

A062731 Sum of divisors of 2*n.

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A224613 a(n) = sigma(6*n).

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A193553 Sum of divisors of 4*n.

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A144613 a(n) = sigma(3*n) = A000203(3*n).

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A283122 a(n) = sigma(8*n).

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A283123 a(n) = sigma(9*n).

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A283118 a(n) = sigma(5*n).

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A144613 a(n) = sigma(3n) = A000203(3n).

A372787 a(n) = tau(7n) = A000005(7n).