A024916 -id:A024916 - cp's OEIS frontend

A259176 Triangle read by rows T(n,k) in which row n lists the odd-indexed terms of n-th row of triangle A237593.

1, 2, 2, 1, 3, 1, 3, 2, 4, 1, 1, 4, 1, 2, 5, 1, 2, 5, 2, 2, 6, 1, 1, 2, 6, 1, 1, 3, 7, 2, 1, 2, 7, 2, 1, 3, 8, 1, 2, 3, 8, 2, 1, 1, 3, 9, 2, 1, 1, 3, 9, 2, 1, 1, 4, 10, 2, 1, 2, 3, 10, 2, 1, 2, 4, 11, 2, 2, 1, 4, 11, 3, 1, 1, 1, 4, 12, 2, 1, 1, 2, 4, 12, 2, 1, 1, 2, 5, 13, 3, 1, 1, 2, 4, 13, 3, 2, 1, 1, 5, 14, 2, 2, 1, 2, 5

Offset: 1

Omar E. Pol, Aug 15 2015

Row n has length A003056(n) hence column k starts in row A000217(k).

Row n is a permutation of the n-th row of A237591 for some n, hence the sequence is a permutation of A237591.

			Written as an irregular triangle the sequence begins:
1;
2;
2, 1;
3, 1;
3, 2;
4, 1, 1;
4, 1, 2;
5, 1, 2;
5, 2, 2;
6, 1, 1, 2;
6, 1, 1, 3;
7, 2, 1, 2;
7, 2, 1, 3;
8, 1, 2, 3;
8, 2, 1, 1, 3;
9, 2, 1, 1, 3;
...
Illustration of initial terms (side view of the pyramid):
Row   _
1    |_|_
2    |_ _|_
3    |_ _|_|_
4    |_ _ _|_|_
5    |_ _ _|_ _|_
6    |_ _ _ _|_|_|_
7    |_ _ _ _|_|_ _|_
8    |_ _ _ _ _|_|_ _|_
9    |_ _ _ _ _|_ _|_ _|_
10   |_ _ _ _ _ _|_|_|_ _|_
11   |_ _ _ _ _ _|_|_|_ _ _|_
12   |_ _ _ _ _ _ _|_ _|_|_ _|_
13   |_ _ _ _ _ _ _|_ _|_|_ _ _|_
14   |_ _ _ _ _ _ _ _|_|_ _|_ _ _|_
15   |_ _ _ _ _ _ _ _|_ _|_|_|_ _ _|_
16   |_ _ _ _ _ _ _ _ _|_ _|_|_|_ _ _|
...
The above structure represents the first 16 levels (starting from the top) of one of the side views of the infinite stepped pyramid described in A245092. For another side view see A259177.
.
Illustration of initial terms (partial front view of the pyramid):
Row                                 _
1                                 _|_|
2                               _|_ _|_
3                             _|_ _| |_|
4                           _|_ _ _| |_|_
5                         _|_ _ _|  _|_ _|
6                       _|_ _ _ _| |_| |_|_
7                     _|_ _ _ _|   |_| |_ _|
8                   _|_ _ _ _ _|  _|_| |_ _|_
9                 _|_ _ _ _ _|   |_ _|_  |_ _|
10              _|_ _ _ _ _ _|   |_| |_| |_ _|_
11            _|_ _ _ _ _ _|    _|_| |_| |_ _ _|
12          _|_ _ _ _ _ _ _|   |_ _| |_|   |_ _|_
13        _|_ _ _ _ _ _ _|     |_ _| |_|_  |_ _ _|
14      _|_ _ _ _ _ _ _ _|    _|_|  _|_ _| |_ _ _|_
15    _|_ _ _ _ _ _ _ _|     |_ _| |_| |_|   |_ _ _|
16   |_ _ _ _ _ _ _ _ _|     |_ _| |_| |_|   |_ _ _|
...
A part of the hidden pattern of the symmetric representation of sigma emerges from the partial front view of the pyramid described in A245092.
For another partial front view see A259177. For the total front view see A237593.

Bisection of A237593.
Row sums give A000027.
For the mirror see A259177 which is another bisection of A237593.
Cf. A000203, A000217, A003056, A024916, A175254, A196020, A236104, A237270, A237271, A237591, A244580, A245092, A249351, A259179, A261350.

(* function f[n,k] and its support functions are defined in A237593 *)
a259176[n_, k_] := f[n, 2*k-1]
TableForm[Table[a259176[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
Flatten[Table[a259176[n, k], {n, 1, 26}, {k, 1, [n]}]] (* sequence data *)
(* Hartmut F. W. Hoft, Mar 06 2017 *)

row(n) = (sqrt(8*n + 1) - 1)\2;
s(n, k) = ceil((n + 1)/k - (k + 1)/2) - ceil((n + 1)/(k + 1) - (k + 2)/2);
T(n, k) = if(k<=row(n), s(n, k), s(n, 2*row(n) + 1 - k));
a259177(n, k) = T(n, 2*k - 1);
for(n=1, 26, for(k=1, row(n), print1(a259177(n, k),", ");); print();)  \\ Indranil Ghosh, Apr 21 2017

from sympy import sqrt
import math
def row(n): return int(math.floor((sqrt(8*n + 1) - 1)/2))
def s(n, k): return int(math.ceil((n + 1)/k - (k + 1)/2)) - int(math.ceil((n + 1)/(k + 1) - (k + 2)/2))
def T(n, k): return s(n, k) if k<=row(n) else s(n, 2*row(n) + 1 - k)
def a259177(n, k): return T(n, 2*k - 1)
for n in range(1, 11): print([a259177(n, k) for k in range(1, row(n) + 1)]) # Indranil Ghosh, Apr 21 2017

Better definition from Omar E. Pol, Apr 26 2021

A326124 a(n) is the sum of all divisors of the first n positive even numbers.

Original entry on oeis.org

3, 10, 22, 37, 55, 83, 107, 138, 177, 219, 255, 315, 357, 413, 485, 548, 602, 693, 753, 843, 939, 1023, 1095, 1219, 1312, 1410, 1530, 1650, 1740, 1908, 2004, 2131, 2275, 2401, 2545, 2740, 2854, 2994, 3162, 3348, 3474, 3698, 3830, 4010, 4244, 4412, 4556, 4808, 4979, 5196, 5412, 5622, 5784, 6064, 6280

Offset: 1

Omar E. Pol, Jun 07 2019

A326123(n)/a(n) converges to 3/5.

a(n) is also the total area of the terraces of the first n even-indexed levels of the stepped pyramid described in A245092.

			For n = 3 the first three positive even numbers are [2, 4, 6] and their divisors are [1, 2], [1, 2, 4], [1, 2, 3, 6] respectively, and the sum of these divisors is 1 + 2 + 1 + 2 + 4 + 1 + 2 + 3 + 6 = 22, so a(3) = 22.

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

Partial sums of A062731.

Cf. A000203, A005843, A024916, A237593, A245092, A299174, A326123.

ListTools:-PartialSums(map(numtheory:-sigma, [seq(i,i=2..200,2)])); # Robert Israel, Jun 12 2019

Accumulate@ DivisorSigma[1, Range[2, 110, 2]] (* Michael De Vlieger, Jun 09 2019 *)

terms(n) = my(s=0, i=0); for(k=1, n-1, if(i>=n, break); s+=sigma(2*k); print1(s, ", "); i++)
/* Print initial 50 terms as follows: */
terms(50) \\ Felix Fröhlich, Jun 08 2019

a(n) = sum(k=1, 2*n, if (!(k%2), sigma(k))); \\ Michel Marcus, Jun 08 2019

from math import isqrt
def A326124(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-3*((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 21 2023

a(n) = A024916(2n) - A326123(n).

a(n) ~ 5 * Pi^2 * n^2 / 24. - Vaclav Kotesovec, Aug 18 2021

A332533 a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

Original entry on oeis.org

1, 4, 15, 92, 790, 9384, 137326, 2397352, 48428487, 1111122360, 28531183329, 810554859732, 25239592620853, 854769763924104, 31278135039463245, 1229782938533709200, 51702516368332126932, 2314494592676172411516, 109912203092257573556274, 5518821052632117898282620

Offset: 1

Ilya Gutkovskiy, Feb 16 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A024916, A268235, A308313, A308814, A319194, A320095.

Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), this sequence (q=n).

A332533:= func< n | (&+[Floor(n/j)*n^(j-1): j in [1..n]]) >;
[A332533(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024

seq(add(n^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023

Table[(1/n) Sum[Floor[n/k] n^k, {k, 1, n}], {n, 1, 20}]
Table[(1/n) Sum[Sum[n^d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
Table[SeriesCoefficient[(1/(1 - x)) Sum[x^k/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 1, 20}]

a(n) = sum(k=1, n, (n\k)*n^k)/n; \\ Michel Marcus, Feb 16 2020

a(n) = sum(k=1, n, sumdiv(k, d, n^(d-1))); \\ Seiichi Manyama, May 29 2021

def A332533(n): return sum((n//j)*n^(j-1) for j in range(1,n+1))
[A332533(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k / (1 - n*x^k).
a(n) = (1/n) * Sum_{k=1..n} Sum_{d|k} n^d.
a(n) ~ n^(n-1). - Vaclav Kotesovec, May 28 2021
a(n) = (1/(n-1)) * Sum_{k=1..n} (n^floor(n/k) - 1), for n>=2. - Ridouane Oudra, Mar 05 2023

A259177 Triangle read by rows T(n,k) in which row n lists the even-indexed terms of n-th row of triangle A237593.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 2, 3, 1, 1, 4, 2, 1, 4, 2, 1, 5, 2, 2, 5, 2, 1, 1, 6, 3, 1, 1, 6, 2, 1, 2, 7, 3, 1, 2, 7, 3, 2, 1, 8, 3, 1, 1, 2, 8, 3, 1, 1, 2, 9, 4, 1, 1, 2, 9, 3, 2, 1, 2, 10, 4, 2, 1, 2, 10, 4, 1, 2, 2, 11, 4, 1, 1, 1, 3, 11, 4, 2, 1, 1, 2, 12, 5, 2, 1, 1, 2, 12, 4, 2, 1, 1, 3, 13, 5, 1, 1, 2, 3, 13, 5, 2, 1, 2, 2, 14

Offset: 1

Omar E. Pol, Aug 15 2015

Row n has length A003056(n) hence column k starts in row A000217(k).

Row n is a permutation of the n-th row of A237591 for some n, hence the sequence is a permutation of A237591.

			Written as an irregular triangle the sequence begins:
1;
2;
1, 2;
1, 3;
2, 3;
1, 1, 4;
2, 1, 4;
2, 1, 5;
2, 2, 5;
2, 1, 1, 6;
3, 1, 1, 6;
2, 1, 2, 7;
3, 1, 2, 7;
3, 2, 1, 8;
3, 1, 1, 2, 8;
3, 1, 1, 2, 9;
...
Illustration of initial terms (side view of the pyramid):
Row                                 _
1                                 _|_|
2                               _|_ _|
3                             _|_|_ _|
4                           _|_|_ _ _|
5                         _|_ _|_ _ _|
6                       _|_|_|_ _ _ _|
7                     _|_ _|_|_ _ _ _|
8                   _|_ _|_|_ _ _ _ _|
9                 _|_ _|_ _|_ _ _ _ _|
10              _|_ _|_|_|_ _ _ _ _ _|
11            _|_ _ _|_|_|_ _ _ _ _ _|
12          _|_ _|_|_ _|_ _ _ _ _ _ _|
13        _|_ _ _|_|_ _|_ _ _ _ _ _ _|
14      _|_ _ _|_ _|_|_ _ _ _ _ _ _ _|
15    _|_ _ _|_|_|_ _|_ _ _ _ _ _ _ _|
16   |_ _ _|_|_|_ _|_ _ _ _ _ _ _ _ _|
...
The above structure represents the first 16 levels (starting from the top) of one of the side views of the infinite stepped pyramid described in A245092. For another side view see A259176.
.
Illustration of initial terms (partial front view of the pyramid):
Row                                 _
1                                  |_|_
2                                 _|_ _|_
3                                |_| |_ _|_
4                               _|_| |_ _ _|_
5                              |_ _|_  |_ _ _|_
6                             _|_| |_| |_ _ _ _|_
7                            |_ _| |_|   |_ _ _ _|_
8                           _|_ _| |_|_  |_ _ _ _ _|_
9                          |_ _|  _|_ _|   |_ _ _ _ _|_
10                        _|_ _| |_| |_|   |_ _ _ _ _ _|_
11                       |_ _ _| |_| |_|_    |_ _ _ _ _ _|_
12                      _|_ _|   |_| |_ _|   |_ _ _ _ _ _ _|_
13                     |_ _ _|  _|_| |_ _|     |_ _ _ _ _ _ _|_
14                    _|_ _ _| |_ _|_  |_|_    |_ _ _ _ _ _ _ _|_
15                   |_ _ _|   |_| |_| |_ _|     |_ _ _ _ _ _ _ _|_
16                   |_ _ _|   |_| |_| |_ _|     |_ _ _ _ _ _ _ _ _|
...
A part of the hidden pattern of the symmetric representation of sigma emerges from the partial front view of the pyramid described in A245092.
For another partial front view see A259176. For the total front view see A237593.

Bisection of A237593.
Row sums give A000027.
Mirror of A259176 which is another bisection of A237593.
Cf. A000203, A000217, A003056, A024916, A175254, A196020, A236104, A237270, A237271, A237591, A244580, A245092, A249351, A259179, A261350.

(* function f[n,k] and its support functions are defined in A237593 *)
a259177[n_, k_] := f[n, 2*k]
TableForm[Table[a259177[n, k], {n, 1, 16}, {k, 1, row[n]}]] (* triangle *)
Flatten[Table[a259177[n, k], {n, 1, 26}, {k, 1, [n]}]] (* sequence data *)
(* Hartmut F. W. Hoft, Mar 06 2017 *)

row(n) = (sqrt(8*n + 1) - 1)\2;
s(n, k) = ceil((n + 1)/k - (k + 1)/2) - ceil((n + 1)/(k + 1) - (k + 2)/2);
T(n, k) = if(k<=row(n), s(n, k), s(n, 2*row(n) + 1 - k));
a259177(n, k) = T(n, 2*k);
for(n=1, 26, for(k=1, row(n), print1(a259177(n, k),", ");); print();) \\ Indranil Ghosh, Apr 21 2017

from sympy import sqrt
import math
def row(n): return int(math.floor((sqrt(8*n + 1) - 1)/2))
def s(n, k): return int(math.ceil((n + 1)/k - (k + 1)/2)) - int(math.ceil((n + 1)/(k + 1) - (k + 2)/2))
def T(n, k): return s(n, k) if k<=row(n) else s(n, 2*row(n) + 1 - k)
def a259177(n, k): return T(n, 2*k)
for n in range(1, 27): print([a259177(n, k) for k in range(1, row(n) + 1)]) # Indranil Ghosh, Apr 21 2017

Better definition from Omar E. Pol, Apr 26 2021

A318742 a(n) = Sum_{k=1..n} floor(n/k)^3.

Original entry on oeis.org

1, 9, 29, 74, 136, 254, 382, 596, 833, 1173, 1505, 2057, 2527, 3209, 3921, 4856, 5674, 6928, 7956, 9474, 10882, 12608, 14128, 16506, 18369, 20797, 23141, 26129, 28567, 32259, 35051, 38963, 42483, 46675, 50435, 55904, 59902, 65156, 70092, 76460

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Richard J. Mathar, Chebyshev approximation of x^m (-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 2.

Cf. A000005, A000203, A024916, A064602, A064603, A064604.

Cf. A006218, A222548, A318743, A318744.

[&+[Floor(n/k)^3:k in [1..n]]: n in [1..40]]; // Marius A. Burtea, Jul 16 2019

Table[Sum[Floor[n/k]^3, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[DivisorSigma[0, k] - 3*DivisorSigma[1, k] + 3*DivisorSigma[2, k], {k, 1, 40}]]

a(n) = sum(k=1, n, (n\k)^3); \\ Michel Marcus, Sep 03 2018

from math import isqrt
def A318742(n): return -(s:=isqrt(n))**4 + sum((q:=n//k)*(3*k*(k-1)+q**2+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 21 2023

a(n) = A006218(n) - 3*A024916(n) + 3*A064602(n).
a(n) ~ zeta(3) * n^3.
G.f.: (1/(1 - x)) * Sum_{k>=1} (3*k*(k - 1) + 1) * x^k/(1 - x^k). - Ilya Gutkovskiy, Jul 16 2019

A353908 Decimal expansion of Pi^2/36.

Original entry on oeis.org

2, 7, 4, 1, 5, 5, 6, 7, 7, 8, 0, 8, 0, 3, 7, 7, 3, 9, 4, 1, 2, 0, 6, 9, 1, 9, 4, 4, 4, 1, 0, 0, 4, 1, 9, 8, 2, 0, 3, 1, 5, 8, 3, 1, 6, 8, 6, 7, 7, 9, 9, 7, 3, 9, 6, 2, 2, 5, 9, 3, 0, 3, 8, 2, 2, 8, 3, 3, 4, 5, 7, 8, 4, 0, 0, 5, 3, 3, 4, 7, 8, 9, 7, 2, 2, 7, 1, 4, 8, 3, 4, 3, 6, 6, 2, 6, 4, 5, 0, 8, 8, 4, 0, 0, 0, 7

Offset: 0

Omar E. Pol, May 10 2022

Ratio between the volume of the stepped pyramid with an infinite number of levels described in A245092 and that of the circumscribed cube (see the first formula).
See also Vaclav Kotesovec's formula (2016) in A175254.
Volume shared by a sphere inscribed in a cube of volume Pi and one of the six pyramids inscribed in the cube. - Omar E. Pol, Sep 01 2024

			0.2741556778080377394120691944410041982031583168677997396225930382283345784...

Index entries for transcendental numbers

Cf. A000203, A000796, A002388, A013661, A019673, A021040, A024916, A072691, A086463, A086729, A091476, A100044, A102753, A175254, A195055, A237271, A237593, A245092.

evalf(Pi^2/36, 121);  # Alois P. Heinz, May 11 2022

RealDigits[Pi^2/36, 10, 100][[1]] (* Amiram Eldar, May 11 2022 *)

PARI
```
Pi^2/36
```
PARI
```
zeta(2)/6
```

Equals lim_{n->oo} A175254(n)/n^3.
Equals A002388/36.
Equals A102753/18.
Equals A195055/12.
Equals A091476/9.
Equals A013661/6.
Equals A100044/4.
Equals A072691/3.
Equals A086463/2.
Equals A086729*2.
Equals A019673^2.
Equals A002388*A021040.
Equals Re(dilog((1+sqrt(3)*i)/2)). - Mohammed Yaseen, Jul 03 2024

A243980 Four times the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

4, 16, 32, 60, 84, 132, 164, 224, 276, 348, 396, 508, 564, 660, 756, 880, 952, 1108, 1188, 1356, 1484, 1628, 1724, 1964, 2088, 2256, 2416, 2640, 2760, 3048, 3176, 3428, 3620, 3836, 4028, 4392, 4544, 4784, 5008, 5368, 5536, 5920, 6096, 6432, 6744, 7032, 7224, 7720

Offset: 1

Omar E. Pol, Jun 18 2014

nonn

Also number of "ON" cells at n-th stage in a structure which looks like a simple 2-dimensional cellular automaton (see example). The structure is formed by the reflection on the four quadrants from the diagram of the symmetry of sigma in the first quadrant after n-th stage, hence the area in each quadrant equals the area of each wedge and equals A024916(n); the sum of all divisors of all positive integers <= n. For more information about the diagram see A237593 and A237270.

			Illustration of the structure after 16 stages (contains 880 ON cells):
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Indranil Ghosh, Table of n, a(n) for n = 1..7342

Partial sums of A239050.
Partial sums give A244050.
Cf. A000203, A000290, A004125, A016742, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934.

Accumulate[4*DivisorSigma[1,Range[50]]] (* Harvey P. Dale, May 13 2018 *)

from math import isqrt
def A243980(n): return -(s:=isqrt(n))**2*(s+1) + sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))<<1 # Chai Wah Wu, Oct 21 2023

a(n) = A016742(n) - 4*A004125(n) = 4*A024916(n).

a(n) = 2*(A006218(n) + A222548(n)) = 2*A327329(n). - Omar E. Pol, Sep 25 2019

A072379 Sum_{k<=n} (sigma(k)^2), where sigma(k) denotes the sum of the divisors of k A000203.

Original entry on oeis.org

1, 10, 26, 75, 111, 255, 319, 544, 713, 1037, 1181, 1965, 2161, 2737, 3313, 4274, 4598, 6119, 6519, 8283, 9307, 10603, 11179, 14779, 15740, 17504, 19104, 22240, 23140, 28324, 29348, 33317, 35621, 38537, 40841, 49122, 50566, 54166, 57302

Offset: 1

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 20 2002

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A024916, A072861.

Cf. A057434, A061502, A074789.

A072379 := proc(n)
    add( numtheory[sigma](k)^2,k=0..n) ;
end proc:
seq(A072379(n),n=1..80) ; # R. J. Mathar, Jul 09 2024

Accumulate[Table[DivisorSigma[1, k]^2, {k, 1, 50}]] (* Vaclav Kotesovec, Sep 10 2018 *)

a(n) = sum(k=1, n, sigma(k)^2) \\ Michel Marcus, Jun 20 2013

Ramanujan's asymptotic formula: (5/6)*Zeta(3)*n^3+O(n^2*log(n)^2)

A072692 Sum of sigma(j) for 1<=j<=10^n, where sigma(j) is the sum of the divisors of j.

Original entry on oeis.org

1, 87, 8299, 823081, 82256014, 8224740835, 822468118437, 82246711794796, 8224670422194237, 822467034112360628, 82246703352400266400, 8224670334323560419029, 822467033425357340138978, 82246703342420509396897774, 8224670334241228180927002517

Offset: 0

Rick L. Shepherd, Jul 02 2002

nonn

			For n=1, the sum of sigma(j) for j<=10 is 1+3+4+7+6+12+8+15+13+18=87, so a(1)=87 (=69+18=A049000(1)+A046915(1)).

P. L. Patodia, Seth Troisi and Hiroaki Yamanouchi, Table of n, a(n) for n = 0..36 (terms a(0)-a(18) by P. L. Patodia and a(19)-a(24) by Seth Troisi)
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, 1747, The Euler Archive, (Eneström Index) E175.
P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916

Compare with A049000. Note that a(n) = A049000(n) + A046915(n).

Cf. A000203 (sigma(n)), A072691 (Pi^2/12), A049000, A046915, A024916, A025281.

for(m=0,10,print1(sum(n=1,k=10^m,n*(k\n)),",")) \\ Improved by M. F. Hasler, Apr 18 2015

A072692(n)=A024916(10^n) \\ This is very efficient, using efficient code of A024916. - M. F. Hasler, Apr 18 2015

[(i, sum([d*(10**i//d) for d in range(1,10**i+1)])) for i in range(8)] # Seth A. Troisi, Jun 27 2010

from math import isqrt
def A072692(n): return -(s:=isqrt(m:=10**n))**2*(s+1)+sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 23 2023

Asymptotic formula: a(n) ~ Pi^2/12 * 10^2n. See A072691 for Pi^2/12. Observe that A025281 also contains that constant in its asymptotic formula.

More terms from P L Patodia (pannalal(AT)usa.net), Jan 11 2008, Jun 25 2008

Corrected by N. J. A. Sloane, Jun 08 2008, following suggestions from Don Reble and David W. Wilson

A244370 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

Original entry on oeis.org

8, 24, 48, 80, 112, 160, 200, 264, 328, 408, 464, 560, 624, 728, 832, 960, 1040, 1184, 1272, 1432, 1576, 1728, 1832, 2024, 2160, 2336, 2512, 2736

Offset: 1

Omar E. Pol, Jun 26 2014

Partial sums of A244371.
If we use toothpicks of length 1/2, so the area of the central square is equal to 1. The total area of the structure after n-th stage is equal to A024916(n), the sum of all divisors of all positive integers <= n, hence the total area of the n-th set of symmetric regions added at n-th stage is equal to sigma(n) = A000203(n), the sum of divisors of n.
If we use toothpicks of length 1, so the number of cells (and the area) of the central square is equal to 4. The number of cells (and the total area) of the structure after n-th stage is equal to 4*A024916(n) = A243980(n), hence the number of cells (and the total area) of the n-th set of symmetric regions added at n-th stage is equal to 4*A000203(n) = A239050(n).

			Illustration of the structure after 16 stages (Contains 960 toothpicks):
.
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244371, A244970, A244971, A245092.

a(n) = 4*A244362(n) = 8*A244360(n).

a(8) corrected and more terms from Omar E. Pol, Oct 18 2014

cp's OEIS Frontend

A259176 Triangle read by rows T(n,k) in which row n lists the odd-indexed terms of n-th row of triangle A237593.

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A326124 a(n) is the sum of all divisors of the first n positive even numbers.

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A332533 a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

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A259177 Triangle read by rows T(n,k) in which row n lists the even-indexed terms of n-th row of triangle A237593.

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A318742 a(n) = Sum_{k=1..n} floor(n/k)^3.

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A353908 Decimal expansion of Pi^2/36.

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