A024916 -id:A024916 - cp's OEIS frontend

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

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

Omar E. Pol, Aug 18 2015

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.

			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.

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

Omar E. Pol, Nov 03 2015

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).

			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).

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.

A294723 a(n) is the total number of vertices after n-th stage in the diagram of the symmetries of sigma described in A236104, with a(0) = 1.

Original entry on oeis.org

1, 4, 7, 11, 16, 20, 27, 31, 38, 45, 53, 57, 66, 70, 78, 89, 100, 104, 115, 119, 130, 142, 150, 154, 167, 176, 184, 196, 211, 215, 230, 234, 249, 261, 269, 280, 297, 301, 309, 321, 338, 342, 359, 363, 379, 398, 406, 410, 429, 440, 459, 471, 487, 491, 510

Offset: 0

Omar E. Pol, Nov 07 2017

nonn

a(n) is also the total number of "hinges" in the "mechanism" where every row of the two-dimensional diagram of the isosceles triangle with n rows described in A237593 is folded in a 90-degree zig-zag, appearing the structure of the stepped pyramid with n levels described in A245092. Note that the diagram described in A236104 is also the top view of the mentioned pyramid. The area of the terraces in the n-th level of the pyramid, starting from the top, equals sigma(n) = A000203(n).
For the construction of the two-dimensional diagram using Dyck paths and for more information about the pyramid see A237593 and A262626.
Note that every line segment of the Dyck paths of the diagram is related to partitions into consecutive parts (see A237591). - Omar E. Pol, Feb 23 2018

			Illustration of initial terms (n = 0..9):
.                                                           _ _ _ _
.                                             _ _ _        |_ _ _  |_
.                                 _ _ _      |_ _ _|       |_ _ _|   |_
.                       _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.               _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.         _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.    .   |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.    1    4      7        11         16           20             27
.
.
.                                               _ _ _ _ _
.                         _ _ _ _ _            |_ _ _ _ _|
.     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
.    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
.    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
.    |_ _ _|   |_ _      |_ _ _|   |_ _  |     |_ _ _|   |_ _  | |
.    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
.    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
.    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
.    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
.
.           31                  38                     45
.
.
Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
|_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.

Robert Price, Table of n, a(n) for n = 0..5000
Omar E. Pol, An infinite stepped pyramid
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Cf. A317109 (number of edges).
Cf. A237590 (number of regions or parts).
Compare with A317293 (analog for the diagram that contains subparts).
Cf. A000203, A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A294847, A296508.

a(n) = A317109(n) - A237590(n) + 1 (Euler's formula). - Omar E. Pol, Jul 21 2018

Terms a(30) and beyond from Robert Price, Jul 31 2018

Example extended for a(7)-a(9) and a(29) by Omar E. Pol, Jul 31 2018

A307159 Partial sums of the bi-unitary divisors sum function: Sum_{k=1..n} bsigma(k), where bsigma is A188999.

Original entry on oeis.org

1, 4, 8, 13, 19, 31, 39, 54, 64, 82, 94, 114, 128, 152, 176, 203, 221, 251, 271, 301, 333, 369, 393, 453, 479, 521, 561, 601, 631, 703, 735, 798, 846, 900, 948, 998, 1036, 1096, 1152, 1242, 1284, 1380, 1424, 1484, 1544, 1616, 1664, 1772, 1822, 1900, 1972, 2042

Offset: 1

Amiram Eldar, Mar 27 2019

nonn

D. Suryanarayana and M. V. Subbarao, Arithmetical functions associated with the biunitary k-ary divisors of an integer, Indian J. Math., Vol. 22 (1980), pp. 281-298.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1, section 4.13.

Cf. A024916, A064609, A188999, A306069, A307042, A307160.

fun[p_,e_] := If[OddQ[e],(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); Accumulate[Array[bsigma, 60]]

a(n) ~ c * n^2, where c = (zeta(2)*zeta(3)/2) * Product_{p}(1 - 2/p^3 + 1/p^4 + 1/p^5 - 1/p^6) (A307160).

A326123 a(n) is the sum of all divisors of the first n odd numbers.

Original entry on oeis.org

1, 5, 11, 19, 32, 44, 58, 82, 100, 120, 152, 176, 207, 247, 277, 309, 357, 405, 443, 499, 541, 585, 663, 711, 768, 840, 894, 966, 1046, 1106, 1168, 1272, 1356, 1424, 1520, 1592, 1666, 1790, 1886, 1966, 2087, 2171, 2279, 2399, 2489, 2601, 2729, 2849, 2947, 3103, 3205, 3309, 3501, 3609, 3719

Offset: 1

Omar E. Pol, Jun 07 2019

a(n)/A326124(n) converges to 3/5.

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

			For n = 3 the first three odd numbers are [1, 3, 5] and their divisors are [1], [1, 3], [1, 5] respectively, and the sum of these divisors is 1 + 1 + 3 + 1 + 5 = 11, so a(3) = 11.

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

Partial sums of A008438.

Cf. A000203, A005408, A024916, A237593, A245092, A326124.

ListTools:-PartialSums(map(numtheory:-sigma, [seq(i,i=1..200,2)])); # Robert Israel, Jun 12 2019

Accumulate@ DivisorSigma[1, Range[1, 109, 2]] (* Michael De Vlieger, Jun 09 2019 *)

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

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

from math import isqrt
def A326123(n): return (-(s:=isqrt(r:=n<<1))**2*(s+1) + sum((q:=r//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) -(t:=isqrt(m:=n>>1))**2*(t+1)+sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))+3*((u:=isqrt(n))**2*(u+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,u+1))>>1) # Chai Wah Wu, Nov 01 2023

a(n) = A024916(2n) - A326124(n).

a(n) ~ Pi^2 * n^2 / 8. - Vaclav Kotesovec, Aug 18 2021

A245093 Triangle read by rows in which row n lists the first n terms of A000203.

Original entry on oeis.org

1, 1, 3, 1, 3, 4, 1, 3, 4, 7, 1, 3, 4, 7, 6, 1, 3, 4, 7, 6, 12, 1, 3, 4, 7, 6, 12, 8, 1, 3, 4, 7, 6, 12, 8, 15, 1, 3, 4, 7, 6, 12, 8, 15, 13, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28

Offset: 1

Omar E. Pol, Jul 15 2014

Reluctant sequence of A000203.
Row sums give A024916.
Has a symmetric representation - for more information see A237270.

			Triangle begins:
1;
1, 3;
1, 3, 4;
1, 3, 4, 7;
1, 3, 4, 7, 6;
1, 3, 4, 7, 6, 12;
1, 3, 4, 7, 6, 12, 8;
1, 3, 4, 7, 6, 12, 8, 15;
1, 3, 4, 7, 6, 12, 8, 15, 13;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A000203, A024916, A027293, A104765, A140207, A221529, A237270, A237593, A245099, A265652.

import Data.List (inits)
a245093 n k = a245093_tabl !! (n-1) !! (k-1)
a245093_row n = a245093_tabl !! (n-1)
a245093_tabl = tail $ inits $ a000203_list
-- Reinhard Zumkeller, Dec 12 2015

T(n,k) = A000203(k), 1<=k<=n.

A327329 Twice the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

2, 8, 16, 30, 42, 66, 82, 112, 138, 174, 198, 254, 282, 330, 378, 440, 476, 554, 594, 678, 742, 814, 862, 982, 1044, 1128, 1208, 1320, 1380, 1524, 1588, 1714, 1810, 1918, 2014, 2196, 2272, 2392, 2504, 2684, 2768, 2960, 3048, 3216, 3372, 3516, 3612, 3860, 3974, 4160, 4304, 4500, 4608, 4848, 4992

Offset: 1

Omar E. Pol, Sep 25 2019

nonn

a(n) has a symmetric representation. Using two opposite quadrants, where in each quadrant there is the Dyck path related to partitions described in the n-th row of triangle A237593, a(n) is the total area (or the total number of cells) of the structure (see the example).

a(n) is also the total area of the horizontal faces in the stepped pyramid with n levels described in A245092 (that is the total area of the terraces plus the area of the base). - Omar E. Pol, Dec 15 2021

			Illustration of a(8) = 112 using a symmetric structure constructed with the Dyck path related to partitions described in the 8th row of triangle A237593.
                           _ _ _ _ _
                          |         |
                          |         |_
                          |           |_ _
                          |               |
                          |     56        |
                          |               |
                          |               |
           _ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _|
          |               |
          |               |
          |               |
          |       56      |
          |_ _            |
              |_          |
                |         |
                |_ _ _ _ _|

Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Partial sums of A074400.

Cf. A001105, A006218, A013661, A024916, A067436, A222548, A236104, A237591, A237593, A243980, A245092, A262626.

Accumulate[2*DivisorSigma[1,Range[60]]] (* Harvey P. Dale, Sep 25 2021 *)

a(n) = 2*sum(k=1, n, sigma(k)); \\ Michel Marcus, Dec 20 2021

from sympy import divisor_sigma
from itertools import accumulate
def f(, n): return  + 2*divisor_sigma(n, 1)
def aupton(terms): return list(accumulate(range(terms+1), f))[1:]
print(aupton(55)) # Michael S. Branicky, Dec 16 2021

from math import isqrt
def A327329(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 22 2023

a(n) = 2*A024916(n).
a(n) = A243980(n)/2.
a(n) = A006218(n) + A222548(n).
a(n) = A001105(n) - A067436(n).
lim_{n->infinity} a(n)/(n^2) = Pi^2/6 = zeta(2) (cf. A013661). - Omar E. Pol, Dec 16 2021

A123229 Triangle read by rows: T(n, m) = n - (n mod m).

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 06 2006

An equivalent definition: Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127093 from the right. That is, T(n,m) = Sum_{j=m..n} A000012(n,j)*A127093(j,m) = Sum_{j=m..n} A127093(j,m) = m*floor(n/m) = m*A010766(n,m). - Gary W. Adamson, Jan 05 2007

The number of parts k in the triangle is A000203(k) hence the sum of parts k is A064987(k). - Omar E. Pol, Jul 05 2014

			Triangle begins:
{1},
{2, 2},
{3, 2, 3},
{4, 4, 3, 4},
{5, 4, 3, 4, 5},
{6, 6, 6, 4, 5, 6},
{7, 6, 6, 4, 5, 6, 7},
{8, 8, 6, 8, 5, 6, 7, 8},
{9, 8, 9, 8, 5, 6, 7, 8, 9},
...

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A074025, A093960.

Cf. also A024916 (row sums), A127093, A127094, A127096, A127097, A127098, A127099, A038040, A000203, A126988, A127013, A127057.

Flat(List([1..10],n->List([1..n],m->n-(n mod m)))); # Muniru A Asiru, Oct 12 2018

seq(seq(n-modp(n,m),m=1..n),n=1..13); # Muniru A Asiru, Oct 12 2018

a = Table[Table[n - Mod[n, m], {m, 1, n}], {n, 1, 20}]; Flatten[a]

for(n=1,9,for(m=1,n,print1(n-n%m", "))) \\ Charles R Greathouse IV, Nov 07 2011

Edited by N. J. A. Sloane, Jul 05 2014 at the suggestion of Omar E. Pol, who observed that A127095 (Gary W. Adamson, with edits by R. J. Mathar) was the same as this sequence.

A244583 a(n) = sum of all divisors of all positive integers <= prime(n).

Original entry on oeis.org

4, 8, 21, 41, 99, 141, 238, 297, 431, 690, 794, 1136, 1384, 1524, 1806, 2304, 2846, 3076, 3699, 4137, 4406, 5128, 5645, 6499, 7755, 8401, 8721, 9393, 9783, 10513, 13280, 14095, 15443, 15871, 18232, 18756, 20320, 21873, 22875, 24604, 26274, 27002, 29982, 30684

Offset: 1

Omar E. Pol, Jun 30 2014

nonn

Limit_{n->oo} a(n)/prime(n)^2 = zeta(2)/2 = Pi^2/12 = A072691 = 0.82246703342.... For example, at n = 2*10^6, the ratio converges to 0.822467033... (+-2 in the last digit with increments on n of +100). If the ratio is calculated with a nonprime for the upper summation limit then the ratio runs slightly larger and converges slower. See formula section of A024916 for the general case. - Richard R. Forberg, Jan 04 2015

This is a subsequence of A024916 therefore a(n) also has a symmetric representation. For more information see A236104, A237593. - Omar E. Pol, Jan 05 2015

Cf. A000040, A000203, A024916, A236104, A237593, A237270, A237271, A244576, A244578.

Partial sums of A358683.

a244583[n_] := Sum[DivisorSigma[1, i], {i, #}] & /@ Prime[Range@n]; a244583[44] (* Michael De Vlieger, Jan 06 2015 *)

a(n) = sum(i=1, prime(n), sigma(i)); \\ Michel Marcus, Sep 29 2014

from math import isqrt
from sympy import prime
def A244583(n): return -(s:=isqrt(p:=prime(n)))**2*(s+1) + sum((q:=p//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 23 2023

a(n) = A024916(A000040(n)).

a(n) = A001248(n) - A050482(n). - Omar E. Pol, Jan 05 2015

More terms from Michel Marcus, Sep 29 2014

A317109 a(n) is the total number of edges after n-th stage in the diagram of the symmetries of sigma described in A236104, with a(0) = 0.

Original entry on oeis.org

0, 4, 8, 14, 20, 26, 34, 40, 48, 58, 68, 74, 84, 90, 100, 114, 126, 132, 144, 150, 162, 178, 188, 194, 208, 220, 230, 246, 262, 268, 284, 290, 306, 322, 332, 346, 364, 370, 380, 396, 414, 420, 438, 444, 462, 484, 494, 500, 520, 534, 556, 572, 590, 596, 616, 636

Offset: 0

Omar E. Pol, Jul 21 2018

nonn

All terms are even numbers.
Note that the two-dimensional diagram is also the top view of the stepped pyramid with n levels described in A245092.
For the construction of the two-dimensional diagram using Dyck paths and for more information about the pyramid see A237593.

			Illustration of initial terms (n = 1..9):
.                                                       _ _ _ _
.                                         _ _ _        |_ _ _  |_
.                             _ _ _      |_ _ _|       |_ _ _|   |_
.                   _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.           _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.     _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.    |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.     4      8        14         20           26             34
.
.                                               _ _ _ _ _
.                         _ _ _ _ _            |_ _ _ _ _|
.     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
.    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
.    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
.    |_ _ _|   |_ _      |_ _ _|   |_ _  |     |_ _ _|   |_ _  | |
.    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
.    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
.    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
.    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
.
.           40                  48                     58
.
.
Illustration of the diagram after 29 stages (contain 268 edges, 215 vertices and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
|_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.

Robert Price, Table of n, a(n) for n = 0..5000

Cf. A294723 (number of vertices).
Cf. A237590 (number of regions or parts).
Compare with A317292 (analog for the diagram that contains subparts).
Cf. A000203, A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A296508.

a(n) = A294723(n) + A237590(n) - 1 (Euler's formula).

More terms and b-file from Robert Price, Jul 31 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

A294723 a(n) is the total number of vertices after n-th stage in the diagram of the symmetries of sigma described in A236104, with a(0) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A307159 Partial sums of the bi-unitary divisors sum function: Sum_{k=1..n} bsigma(k), where bsigma is A188999.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A326123 a(n) is the sum of all divisors of the first n odd numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

A245093 Triangle read by rows in which row n lists the first n terms of A000203.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

A327329 Twice the sum of all divisors of all positive integers <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Formula

A123229 Triangle read by rows: T(n, m) = n - (n mod m).

Views

Author

Keywords

Comments