cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Omar E. Pol, May 10 2022

Keywords

Comments

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

Examples

			0.2741556778080377394120691944410041982031583168677997396225930382283345784...

Links

Crossrefs

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

Programs

  • Maple
    evalf(Pi^2/36, 121);  # Alois P. Heinz, May 11 2022
  • Mathematica
    RealDigits[Pi^2/36, 10, 100][[1]] (* Amiram Eldar, May 11 2022 *)
  • PARI
    Pi^2/36
  • PARI
    zeta(2)/6

Formula

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

Views

Author

Omar E. Pol, Jun 18 2014

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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.

Programs

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

Formula

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

A261350 Triangle read by rows T(n,k) which is the mirror of A237591.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Aug 18 2015

Keywords

Comments

Row n has length A003056(n) hence column k starts in row A000217(k).
Row sums give A000027.
Right border gives A008619, n >= 1.
n is an odd prime if and only if T(n,r-1) = 1 + T(n-1,r-1) and T(n,k) = T(n-1,k) for the rest of the values of k, where r = A003056(n) is the number of elements in row n.
T(n,k) is also the length of the k-th segment in a zig-zag path on the first quadrant of the square grid, connecting the point (m, m) with the point (0, n), ending with a segment in horizontal direction, where m = A240542(n). The area of the polygon defined by the y-axis, this zig-zag path and the diagonal [(0, 0), (m, m)], is equal to A024916(n)/2, one half of the sum of all divisors of all positive integers <= n. Therefore the reflected polygon, which is adjacent to the x-axis, with the zig-zag path connecting the point (n, 0) with the point (m, m), has the same property. And so on for each octant in the four quadrants.
For the representation of A024916 and A000203 we use two octants, for example: the first octant and the second octant, or the 6th octant and the 7th octant, etc., see A237593.
The elements of the n-th row of A237591 together with the elements of the n-th row of this sequence give the n-th row of A237593.
The connection between A196020 and A237271 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> this sequence --> A237593 --> A239660 --> A237270 --> A237271.
T(n,k) is also the area (or the number of cells) of the k-th vertical side at the n-th level (starting from the top) in the right part of the front view of the stepped pyramid described in A245092, see Example section.

Examples

			Triangle begins:
Row
1                       1;
2                       2;
3                    1, 2;
4                    1, 3;
5                    2, 3;
6                 1, 1, 4;
7                 1, 2, 4;
8                 1, 2, 5;
9                 2, 2, 5;
10             1, 1, 2, 6;
11             1, 1, 3, 6;
12             1, 2, 2, 7;
13             1, 2, 3, 7;
14             2, 1, 3, 8;
15          1, 1, 2, 3, 8;
16          1, 1, 2, 3, 9;
17          1, 1, 2, 4, 9;
18          1, 2, 2, 3, 10;
19          1, 2, 2, 4, 10;
20          2, 1, 2, 4, 11;
21       1, 1, 1, 3, 4, 11;
22       1, 1, 2, 2, 4, 12;
23       1, 1, 2, 2, 5, 12;
24       1, 1, 2, 3, 4, 13;
25       1, 2, 1, 3, 5, 13;
26       1, 2, 2, 2, 5, 14;
...
Illustration of initial terms:
Row      _
1       |1|_
2       |_ 2|_
3       |1|  2|_
4       |1|_   3|_
5       |_ 2|    3|_
6       |1|1|_     4|_
7       |1|  2|      4|_
8       |1|_ 2|_       5|_
9       |_ 2|  2|        5|_
10      |1|1|  2|_         6|_
11      |1|1|_   3|          6|_
12      |1|  2|  2|_           7|_
13      |1|_ 2|    3|            7|_
14      |_ 2|1|_   3|_             8|_
15      |1|1|  2|    3|              8|_
16      |1|1|  2|    3|_               9|_
17      |1|1|_ 2|_     4|                9|_
18      |1|  2|  2|    3|_                10|_
19      |1|_ 2|  2|      4|                 10|_
20      |_ 2|1|  2|_     4|_                  11|_
21      |1|1|1|_   3|      4|                   11|_
22      |1|1|  2|  2|      4|_                    12|_
23      |1|1|  2|  2|_       5|                     12|_
24      |1|1|_ 2|    3|      4|_                      13|_
25      |1|  2|1|_   3|        5|                       13|_
26      |1|  2|  2|  2|        5|                         14|
...
Also the diagram represents the right part of the front view of the pyramid described in A245092. For the other half front view see A237591. For more information about the pyramid and the symmetric representation of sigma see A237593.

Crossrefs

Cf. A000203, A000217, A003056, A008016, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244580, A245092, A249351, A259176, A259177, A259179.

A262612 Triangle read by rows T(n,k) in which column k lists the partial sums of the k-th column of triangle A236104.

Original entry on oeis.org

1, 5, 14, 1, 30, 2, 55, 6, 91, 10, 1, 140, 19, 2, 204, 28, 3, 285, 44, 7, 385, 60, 11, 1, 506, 85, 15, 2, 650, 110, 24, 3, 819, 146, 33, 4, 1015, 182, 42, 8, 1240, 231, 58, 12, 1, 1496, 280, 74, 16, 2, 1785, 344, 90, 20, 3, 2109, 408, 115, 29, 4, 2470, 489, 140, 38, 5, 2870, 570, 165, 47, 9, 3311, 670, 201, 56, 13, 1
Offset: 1

Views

Author

Omar E. Pol, Nov 03 2015

Keywords

Comments

Alternating sum of row n equals A175254(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A175254(n), which is also the volume (or the total number of units cubes) in the first n levels of the stepped pyramid described in A245092.
Row n has length A003056(n) hence the first element of column k is in row A000217(k).

Examples

			Triangle begins:
     1;
     5;
    14,    1;
    30,    2;
    55,    6;
    91,   10,    1;
   140,   19,    2;
   204,   28,    3;
   285,   44,    7;
   385,   60,   11,    1;
   506,   85,   15,    2;
   650,  110,   24,    3;
   819,  146,   33,    4;
  1015,  182,   42,    8;
  1240,  231,   58,   12,    1;
  1496,  280,   74,   16,    2;
  1785,  344,   90,   20,    3;
  2109,  408,  115,   29,    4;
  2470,  489,  140,   38,    5;
  2870,  570,  165,   47,    9;
  3311,  670,  201,   56,   13,    1;
  3795,  770,  237,   72,   17,    2;
  4324,  891,  273,   88,   21,    3;
  4900, 1012,  322,  104,   25,    4;
  ...
For n = 6 we have that A175254(6) = [1] + [1 + 3] + [1 + 3 + 4] + [1 + 3 + 4 + 7] + [1 + 3 + 4 + 7 + 6] + [1 + 3 + 4 + 7 + 6 + 12] = 1 + 4 + 8 + 15 + 21 + 33 = 82. On the other hand the alternating sum of the 6th row of the triangle is 91 - 10 + 1 = 82, equaling A175254(6).

Crossrefs

Cf. A000203, A000217, A003056, A024916, A175254, A196020, A235791, A236104, A237048, A237591, A237593, A237270, A237271, A245092, A261699, A262626.
Column 1 gives A000330, n >= 1. Column 2 is A005993. It appears that column 3 is A092353.

A072481 a(n) = Sum_{k=1..n} Sum_{d=1..k} (k mod d).

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 9, 17, 25, 37, 50, 72, 89, 117, 148, 184, 220, 271, 318, 382, 443, 513, 590, 688, 773, 876, 988, 1113, 1237, 1388, 1526, 1693, 1860, 2044, 2241, 2459, 2657, 2890, 3138, 3407, 3665, 3962, 4246, 4571, 4899, 5238, 5596, 5999, 6373, 6787, 7207
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 02 2002

Keywords

Comments

Previous name was: Sums of sums of remainders when dividing n by k, 0
Partial sums of A004125.

Links

Crossrefs

Cf. A224923, A224924.

Programs

  • Maple
    N:= 200: # to get a(0) to a(N)
S:= series(add(k*x^(2*k)/(1-x^k),k=1..floor(N/2))/(1-x)^2, x, N+1):
seq((n^3-n)/6 - coeff(S,x,n), n=0..N); # Robert Israel, Aug 13 2015
  • Mathematica
    a[n_] := n(n+1)(2n+1)/6 - Sum[DivisorSigma[1, k] (n-k+1), {k, 1, n}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 08 2019, after Omar E. Pol *)
  • PARI
    a(n) = sum(k=1, n, sum(d=1, k, k % d)); \\ Michel Marcus, Feb 11 2014
  • Python
    for n in range(99):
    s = 0
    for k in range(1,n+1):
      for d in range(1,k+1):
        s += k % d
    print(str(s), end=',')
  • Python
    from math import isqrt
def A072481(n): return (n*(n+1)*((n<<1)+1)-((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1)-sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 22 2023

Formula

a(n) = Sum_{k=1..n} Sum_{d=1..k}(k mod d).
a(n) = A000330(n) - A175254(n), n >= 1. - Omar E. Pol, Aug 12 2015
G.f.: x^2/(1-x)^4 - (1-x)^(-2) * Sum_{k>=1} k*x^(2*k)/(1-x^k). - Robert Israel, Aug 13 2015
a(n) ~ (1 - Pi^2/12)*n^3/3. - Vaclav Kotesovec, Sep 25 2016

Extensions

New name and a(0) from Alex Ratushnyak, Feb 10 2014

A294017 Partial sums of A294016.

Original entry on oeis.org

1, 5, 12, 26, 43, 73, 106, 154, 211, 285, 362, 472, 585, 719, 872, 1056, 1243, 1473, 1706, 1984, 2285, 2615, 2948, 3354, 3773, 4225, 4704, 5240, 5779, 6403, 7030, 7720, 8441, 9203, 9992, 10892, 11795, 12743, 13726, 14810, 15897, 17093, 18292, 19572, 20919, 22319, 23722, 25278, 26851, 28511, 30214, 32010, 33809
Offset: 1

Views

Author

Omar E. Pol, Oct 22 2017

Keywords

Comments

a(n) is also the volume of another version of the pyramid with n levels (starting from the top) described in A245092. Both pyramids have the same top view and the same front view, but in this pyramid the volume in the n-th level is equal to A294016(n) instead of A024916(n).

Crossrefs

Cf. A000203, A072481, A175254, A237593, A245092, A294015, A294016.

Programs

  • Python
    from math import isqrt
def A294017(n): return (((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1,s+1)))//3-n*(n+1)*((n<<1)+1)//6 # Chai Wah Wu, Oct 22 2023

Formula

a(n) = A175254(n) - A072481(n).

A340584 Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.

Original entry on oeis.org

1, 4, 7, 1, 11, 3, 1, 13, 4, 3, 1, 18, 7, 4, 3, 1, 20, 6, 7, 4, 3, 1, 23, 12, 6, 7, 4, 3, 1, 28, 8, 12, 6, 7, 4, 3, 1, 31, 15, 8, 12, 6, 7, 4, 3, 1, 30, 13, 15, 8, 12, 6, 7, 4, 3, 1, 40, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 42, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 38, 28, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1
Offset: 1

Views

Author

Omar E. Pol, Jan 12 2021

Keywords

Comments

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower (a polycube) described in A221529 which has A000041(n-1) levels in total. The terraces of the polycube are the symmetric representation of sigma. The terraces are in the levels that are the partition numbers A000041 starting from the base. Note that for n >= 2 there are n - 1 terraces because the first terrace of the tower is formed by two symmetric representations of sigma in the same level. The volume (or the number of cubes) equals A066186(n), the sum of all parts of all partitions of n. The volume is also the sum of all divisors of all terms of the first n rows of A336811. That is due to the correspondence between divisors and partitions (cf. A336811). The growth of the volume (A066186) represents the convolution of A000203 and A000041.

Examples

			Triangle begins:
   1;
   4;
   7,  1;
  11,  3,  1;
  13,  4,  3,  1;
  18,  7,  4,  3,  1;
  20,  6,  7,  4,  3,  1;
  23, 12,  6,  7,  4,  3,  1;
  28,  8, 12,  6,  7,  4,  3,  1;
  31, 15,  8, 12,  6,  7,  4,  3,  1;
  30, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  40, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  42, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  38, 28, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the first term of row 7 is T(7,1) = 20. The other terms in row 7 are the first five terms of A000203 in reverse order, that is [6, 7, 4, 3, 1] so the 7th row of the triangle is [20, 6, 7, 4, 3, 1].
From _Omar E. Pol_, Jul 11 2021: (Start)
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due to the first two partition numbers A000041 are [1, 1]), so T(7,1) = sigma(7) + sigma(6) = 8 + 12 = 20. (End)

Crossrefs

The length of row n is A028310(n-1).
Row sums give A024916.
Column 1 gives 1 together with A092403.
Other columns give A000203.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A346533 (mirror).
Cf. A000041, A221529, A237270, A237593, A245092, A272172 (similar), A336811, A338156, A339106, A340035.

Programs

  • Mathematica
    Table[If[n <= 2, {Total@ #}, Prepend[#2, Total@ #1] & @@ TakeDrop[#, 2]] &@ DivisorSigma[1, Range[n, 1, -1]], {n, 14}] // Flatten (* Michael De Vlieger, Jan 13 2021 *)

A256533 Product of n and the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

1, 8, 24, 60, 105, 198, 287, 448, 621, 870, 1089, 1524, 1833, 2310, 2835, 3520, 4046, 4986, 5643, 6780, 7791, 8954, 9913, 11784, 13050, 14664, 16308, 18480, 20010, 22860, 24614, 27424, 29865, 32606, 35245, 39528, 42032, 45448, 48828, 53680, 56744, 62160, 65532, 70752, 75870, 80868, 84882, 92640, 97363, 104000
Offset: 1

Views

Author

Omar E. Pol, May 02 2015

Keywords

Comments

a(n) is also sum of the volumes (or the total number of unit cubes) from two complementary polycubes: the irregular staircase after n-th stage described in A244580, and the irregular stepped pyramid after (n-1)st stage described in A245092. Note that in both structures the horizontal area in the n-th level is also the symmetric representation of sigma(n). This comment is represented by the third formula.

Examples

			For n = 3; a(3) = 3 * 8 = 19 + 5 = 24.

Links

Crossrefs

Cf. A000203, A000578, A024916, A064987, A143128, A175254, A196020, A236104, A237270, A237271, A237593, A244580, A245092, A245100, A256532, A332264.

Programs

  • Mathematica
    a[n_]:=n*Apply[Plus,Flatten[Divisors[Range[n]]]]; Array[a,50] (* Ivan N. Ianakiev, May 03 2015 *)
nxt[{n_,sd_,a_}]:=Module[{k=(n+1)*(DivisorSigma[1,n+1]+sd)},{n+1,sd+DivisorSigma[ 1,n+1],k}]; NestList[ nxt,{1,1,1},50][[;;,3]] (* Harvey P. Dale, Jun 12 2023 *)
  • PARI
    a(n) = n*sum(k=1, n, n\k*k); \\ Michel Marcus, Apr 29 2020
  • Python
    def A256533(n):
    s=0
    for k in range(1, n+1):
        s+=n%k
    return (n**3)-(s*n) # Indranil Ghosh, Feb 13 2017
  • Python
    from math import isqrt
def A256533(n): return n*(-(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 22 2023

Formula

a(n) = n*A024916(n).
a(n) = n^3 - A256532(n).
a(n) = A143128(n) + A175254(n-1), n > 1.
a(n) = A332264(n) + A175254(n). - Omar E. Pol, Apr 29 2020

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 4, 9, 8, 7, 5, 12, 12, 14, 6, 6, 15, 16, 21, 12, 12, 7, 18, 20, 28, 18, 24, 8, 8, 21, 24, 35, 24, 36, 16, 15, 9, 24, 28, 42, 30, 48, 24, 30, 13, 10, 27, 32, 49, 36, 60, 32, 45, 26, 18, 11, 30, 36, 56, 42, 72, 40, 60, 39, 36, 12, 12, 33, 40, 63, 48, 84, 48, 75, 52, 54, 24, 28, 13, 36, 44, 70, 54, 96, 56, 90, 65, 72, 36, 56, 14
Offset: 1

Views

Author

Omar E. Pol, Oct 02 2016

Keywords

Comments

Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.
In this case the sequence S is A000203.
The n-th diagonal of this triangle gives n times A000203.
The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.
T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.
In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.
The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.
For an illustration of the pyramid, see the Links section.
The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.
Column k lists the positive multiples of sigma(k).
The k-th term in the n-th diagonal is equal to n*sigma(k).
Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.

Examples

			Triangle begins:
1;
2,  3;
3,  6,  4;
4,  9,  8,  7;
5,  12, 12, 14, 6;
6,  15, 16, 21, 12, 12;
7,  18, 20, 28, 18, 24,  8;
8,  21, 24, 35, 24, 36,  16, 15;
9,  24, 28, 42, 30, 48,  24, 30,  13;
10, 27, 32, 49, 36, 60,  32, 45,  26,  18;
11, 30, 36, 56, 42, 72,  40, 60,  39,  36,  12;
12, 33, 40, 63, 48, 84,  48, 75,  52,  54,  24, 28;
13, 36, 44, 70, 54, 96,  56, 90,  65,  72,  36, 56,  14;
14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
...
For n = 16 and k = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, and 16 - 10 + 1 = 7, and 7*18 = 126, so T(16,10) = 126.
On the other hand, the symmetric representation of sigma(10) has two parts of 9 cells, giving a total of 18 cells. In the stepped pyramid described in A245092, with 16 levels, there are 16 - 10 + 1 = 7 cubes exactly below every cell of the symmetric representation of sigma(10) up the base of pyramid; hence the total numbers of cubes exactly below the terraces of the 10th level (starting from the top) up the base of the pyramid is equal to 7*18 = 126. So T(16,10) = 126.
The sum of the 16th row of the triangle is 16 + 45 + 56 + 91 + 72 + 132 + 80 + 135 + 104 + 126 + 72 + 140 + 56 + 72 + 48 + 31 = A175254(16) = 1276, equaling the volume (also the number of cubes) of the stepped pyramid with 16 levels described in A245092 (see Links section).

Links

Crossrefs

Row sums of triangle give A175254.
Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.
Column 1 is A000027.
Initial zeros should be omitted in the following sequences related to the columns of triangle:
Columns 2-5: A008585, A008586, A008589, A008588.
Columns 6 and 11: A008594.
Columns 7-9: A008590, A008597, A008595.
Columns 10 and 17: A008600.
Columns 12-13: A135628, A008596.
Columns 14, 15 and 23: A008606.
Columns 16 and 25: A135631.
(There are many other OEIS sequences that are also columns of this triangle.)
Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.

Formula

T(n,k) = (n-k+1) * A000203(k).
T(n,k) = A004736(n,k) * A245093(n,k).

A346533 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.

Original entry on oeis.org

1, 4, 1, 7, 1, 3, 11, 1, 3, 4, 13, 1, 3, 4, 7, 18, 1, 3, 4, 7, 6, 20, 1, 3, 4, 7, 6, 12, 23, 1, 3, 4, 7, 6, 12, 8, 28, 1, 3, 4, 7, 6, 12, 8, 15, 31, 1, 3, 4, 7, 6, 12, 8, 15, 13, 30, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 40, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 42, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 38
Offset: 1

Views

Author

Omar E. Pol, Jul 22 2021

Keywords

Comments

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the top of the symmetric tower (a polycube) described in A221529.
The height of the tower equals A000041(n-1).
The terraces of the tower are the symmetric representation of sigma.
The terraces are in the levels that are the partition numbers A000041 starting from the base.
Note that for n >= 2 there are n - 1 terraces because the lower terrace of the tower is formed by two symmetric representations of sigma in the same level.

Examples

			Triangle begins:
  1;
  4;
  1, 7;
  1, 3, 11;
  1, 3,  4, 13;
  1, 3,  4,  7, 18;
  1, 3,  4,  7,  6, 20;
  1, 3,  4,  7,  6, 12, 23;
  1, 3,  4,  7,  6, 12,  8, 28;
  1, 3,  4,  7,  6, 12,  8, 15, 31;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 30;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 40;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 42;
  1, 3,  4,  7,  6, 12,  8, 15, 13, 18, 12, 28, 38;
  ...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the last term of row 7 is T(7,6) = 20. The other terms in row 7 are the first five terms of A000203, so the 7th row of the triangle is [1, 3, 4, 7, 6, 20].
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due the first two partition numbers A000041 are [1, 1]), so T(7,6) = sigma(7) + sigma(6) = 8 + 12 = 20.
.
Illustration of initial terms:
   Row 1    Row 2      Row 3      Row 4        Row 5          Row 6
.
    1        4         1 7        1 3 11       1 3 4 13       1 3 4 7 18
.   _        _ _       _ _ _      _ _ _ _      _ _ _ _ _      _ _ _ _ _ _
   |_|      |   |     |_|   |    |_| |   |    |_| | |   |    |_| | | |   |
            |_ _|     |    _|    |_ _|   |    |_ _|_|   |    |_ _|_| |   |
                      |_ _|      |      _|    |_ _|  _ _|    |_ _|  _|   |
                                 |_ _ _|      |     |        |_ _ _|    _|
                                              |_ _ _|        |        _|
                                                             |_ _ _ _|
.

Links

Crossrefs

Mirror of A340584.
The length of row n is A028310(n-1).
Row sums give A024916.
Leading diagonal gives A092403.
Other diagonals give A000203.
Companion of A346562.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A000041, A221529, A237270, A237593, A245092, A245093 (similar), A336811, A336812, A338156, A339106, A340035.

Programs

  • Mathematica
    A346533row[n_]:=If[n==1,{1},Join[DivisorSigma[1,Range[n-2]],{Total[DivisorSigma[1,{n-1,n}]]}]];Array[A346533row,15] (* Paolo Xausa, Oct 23 2023 *)
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