A239931 -id:A239931 - cp's OEIS frontend

A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.

1, 2, 3, 4, 5, 9, 6, 7, 15, 21, 8, 10, 25, 27, 63, 12, 11, 35, 33, 81, 147, 16, 13, 45, 39, 99, 171, 357, 18, 14, 49, 51, 117, 189, 399, 903, 20, 17, 50, 55, 153, 207, 441, 987, 2499, 24, 19, 70, 57, 165, 243, 483, 1029, 2709, 6069, 28, 22, 77, 65, 195, 261, 513, 1113

Offset: 1

Omar E. Pol, Apr 06 2014

This is a permutation of the positive integers.
All odd primes are in column 2 (together with some even composite numbers) because the symmetric representation of sigma(prime(i)) is [m, m], where m = (1 + prime(i))/2, for i >= 2.
The union of all odd-indexed columns gives A071562, the positive integers that have middle divisors. The union of all even-indexed columns gives A071561, the positive integers without middle divisors. - Omar E. Pol, Oct 01 2018
Each column in the table of A357581 is a subsequence of the respective column in the table of this sequence; however, the first row in the table of A357581 is not a subsequence of the first row in the table of this sequence. - Hartmut F. W. Hoft, Oct 04 2022
Conjecture: T(n,k) is the n-th positive integer with k 2-dense sublists of divisors. - Omar E. Pol, Aug 25 2025

			Array begins:
   1,  3,  9, 21,  63, 147, 357,  903, 2499, 6069, ...
   2,  5, 15, 27,  81, 171, 399,  987, 2709, 6321, ...
   4,  7, 25, 33,  99, 189, 441, 1029, 2793, 6325, ...
   6, 10, 35, 39, 117, 207, 483, 1113, 2961, 6783, ...
   8, 11, 45, 51, 153, 243, 513, 1197, 3025, 6875, ...
  12, 13, 49, 55, 165, 261, 567, 1239, 3087, 6909, ...
  16, 14, 50, 57, 195, 275, 609, 1265, 3249, 7011, ...
  18, 17, 70, 65, 231, 279, 621, 1281, 3339, 7203, ...
  20, 19, 77, 69, 255, 297, 651, 1375, 3381, 7353, ...
  24, 22, 91, 75, 273, 333, 729, 1407, 3591, 7581, ...
  ...
[Lower right hand triangle of array completed by _Hartmut F. W. Hoft_, Oct 04 2022]

Index entries for sequences that are permutations of the natural numbers

For programs see A237271 and A237593.
Row 1 is A239663.
Column 1 is A174973.
Columns 2..7: A239929, A279102, A280107, A320066, A320511, A357775.
For more information see A239663 and A239665.
Cf. A000203, A006254, A065091, A067742, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A239660, A239929, A239931-A239934, A245092, A262626, A319529, A319796, A319801, A319802.
Cf. A341969, A341970, A341971, A357581.

(* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
widthTable[n_, {r_, c_}] := Module[{k, list=Table[{}, c], parts}, For[k=1, k<=n, k++, parts=partsSRS[k]; If[parts<=c&&Length[list[[parts]]]=1, j--, vec[[PolygonalNumber[i+j-2]+j]]=arr[[i, j]]]]; vec]
a240062T[n_, r_] := TableForm[widthTable[n, {r, r}]]
a240062[6069, 10] (* data *)
a240062T[7581, 10] (* 10 X 10 array - Hartmut F. W. Hoft, Oct 04 2022 *)

a(n) > 128 from Michel Marcus, Apr 08 2014

A239932 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-2).

Original entry on oeis.org

3, 12, 9, 9, 12, 12, 39, 18, 18, 21, 21, 72, 27, 27, 30, 30, 96, 36, 36, 39, 15, 39, 120, 45, 45, 48, 48, 144, 54, 36, 54, 57, 57, 84, 84, 63, 63, 66, 66, 234, 72, 72, 75, 21, 75, 108, 108, 81, 81, 84, 48, 84, 120, 120, 90, 90, 93, 93, 312

Offset: 1

Omar E. Pol, Mar 29 2014

Row n is a palindromic composition of sigma(4n-2).
Row n is also the row 4n-2 of A237270.
Row n has length A237271(4n-2).
Row sums give A239052.
Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the second quadrant of the spiral described in A239660, see example.
For the parts of the symmetric representation of sigma(4n-3), see A239931.
For the parts of the symmetric representation of sigma(4n-1), see A239933.
For the parts of the symmetric representation of sigma(4n), see A239934.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

			The irregular triangle begins:
3;
12;
9, 9;
12, 12;
39;
18, 18;
21, 21;
72;
27, 27;
30, 30;
96;
36, 36;
39, 15, 39;
120;
45, 45;
48, 48;
...
Illustration of initial terms in the second quadrant of the spiral described in A239660:
.                                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                                | |
.                                | |
.                                | |  _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                           _ _ _| | |  _ _ _ _ _ _ _ _ _ _ _ _ _|
.                          |       | | |
.                       _ _|  _ _ _| | |
.                  72 _|     |       | |  _ _ _ _ _ _ _ _ _ _ _ _
.                   _|      _| 21 _ _| | |  _ _ _ _ _ _ _ _ _ _ _|
.                  |      _|     |_ _ _| | |
.               _ _|    _|    _ _|       | |
.              |    _ _|    _|     18 _ _| |  _ _ _ _ _ _ _ _ _ _
.              |   |       |         |_ _ _| |  _ _ _ _ _ _ _ _ _|
.     _ _ _ _ _|   | 21 _ _|        _|       | |
.    |  _ _ _ _ _ _|   | |        _|      _ _| |
.    | |      _ _ _ _ _| | 18 _ _|       |     |  _ _ _ _ _ _ _ _
.    | |     |  _ _ _ _ _|   | |     39 _|  _ _| |  _ _ _ _ _ _ _|
.    | |     | |      _ _ _ _| |    _ _|  _|     | |
.    | |     | |     |  _ _ _ _|   |    _|   12 _| |
.    | |     | |     | |      _ _ _|   |       |_ _|  _ _ _ _ _ _
.    | |     | |     | |     |  _ _ _ _| 12 _ _|     |  _ _ _ _ _|
.    | |     | |     | |     | |      _ _ _| |    9 _| |
.    | |     | |     | |     | |     |  _ _ _|  9 _|_ _|
.    | |     | |     | |     | |     | |      _ _| |      _ _ _ _
.    | |     | |     | |     | |     | |     |  _ _| 12 _|  _ _ _|
.    | |     | |     | |     | |     | |     | |      _|   |
.    | |     | |     | |     | |     | |     | |     |  _ _|
.    | |     | |     | |     | |     | |     | |     | |    3 _ _
.    | |     | |     | |     | |     | |     | |     | |     |  _|
.    |_|     |_|     |_|     |_|     |_|     |_|     |_|     |_|
.
For n = 7 we have that 4*7-2 = 26 and the 26th row of A237593 is [14, 5, 2, 2, 2, 1, 1, 2, 2, 2, 5, 14] and the 25th row of A237593 is [13, 5, 3, 1, 2, 1, 1, 2, 1, 3, 5, 13] therefore between both Dyck paths there are two regions (or parts) of sizes [21, 21], so row 7 is [21, 21].
The sum of divisors of 26 is 1 + 2 + 13 + 26 = A000203(26) = 42. On the other hand the sum of the parts of the symmetric representation of sigma(26) is 21 + 21 = 42, equaling the sum of divisors of 26.

Cf. A000203, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239052, A239660, A239931, A239933, A239934, A244050, A245092, A262626.

A239933 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-1).

Original entry on oeis.org

2, 2, 4, 4, 6, 6, 8, 8, 8, 10, 10, 12, 12, 14, 6, 6, 14, 16, 16, 18, 12, 18, 20, 8, 8, 20, 22, 22, 24, 24, 26, 10, 10, 26, 28, 8, 8, 28, 30, 30, 32, 12, 16, 12, 32, 34, 34, 36, 36, 38, 24, 24, 38, 40, 40, 42, 42, 44, 16, 16, 44, 46, 20, 46, 48, 12, 12, 48, 50, 18, 20, 18, 50, 52, 52, 54, 54, 56, 20, 20, 56, 58, 14, 14, 58, 60, 12, 12, 60, 62, 22, 22, 62, 64, 64

Offset: 1

Omar E. Pol, Mar 29 2014

Row n is a palindromic composition of sigma(4n-1).
Row n is also the row 4n-1 of A237270.
Row n has length A237271(4n-1).
Row sums give A239053.
Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the third quadrant of the spiral described in A239660, see example.
For the parts of the symmetric representation of sigma(4n-3), see A239931.
For the parts of the symmetric representation of sigma(4n-2), see A239932.
For the parts of the symmetric representation of sigma(4n), see A239934.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

			The irregular triangle begins:
2, 2;
4, 4;
6, 6;
8, 8, 8;
10, 10;
12, 12;
14, 6, 6, 14;
16, 16;
18, 12, 18;
20, 8, 8, 20;
22, 22;
24, 24;
26, 10, 10, 26;
28, 8, 8, 28;
30, 30;
32, 12, 16, 12, 32;
...
Illustration of initial terms in the third quadrant of the spiral described in A239660:
.     _       _       _       _       _       _       _       _
.    | |     | |     | |     | |     | |     | |     | |     | |
.    | |     | |     | |     | |     | |     | |     | |     |_|_ _
.    | |     | |     | |     | |     | |     | |     | |    2  |_ _|
.    | |     | |     | |     | |     | |     | |     |_|_     2
.    | |     | |     | |     | |     | |     | |    4    |_
.    | |     | |     | |     | |     | |     |_|_ _        |_ _ _ _
.    | |     | |     | |     | |     | |    6      |_      |_ _ _ _|
.    | |     | |     | |     | |     |_|_ _ _        |_   4
.    | |     | |     | |     | |    8      | |_ _      |
.    | |     | |     | |     |_|_ _ _      |_    |     |_ _ _ _ _ _
.    | |     | |     | |   10        |       |_  |_    |_ _ _ _ _ _|
.    | |     | |     |_|_ _ _ _      |_ _   8  |_ _|  6
.    | |     | |   12          |         |_        |
.    | |     |_|_ _ _ _ _      |_ _        |       |_ _ _ _ _ _ _ _
.    | |   14          | |         |_      |_ _    |_ _ _ _ _ _ _ _|
.    |_|_ _ _ _ _      | |_ _        |_        |  8
.  16            |     |_ _  |         |       |
.                |         |_|_        |_ _    |_ _ _ _ _ _ _ _ _ _
.                |_ _     6    |_ _        |   |_ _ _ _ _ _ _ _ _ _|
.                    |         |_  |       | 10
.                    |_       6  | |_ _    |
.                      |_        |_ _ _|   |_ _ _ _ _ _ _ _ _ _ _ _
.                        |_ _          |   |_ _ _ _ _ _ _ _ _ _ _ _|
.                            |         | 12
.                            |_ _ _    |
.                                  |   |_ _ _ _ _ _ _ _ _ _ _ _ _ _
.                                  |   |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                                  | 14
.                                  |
.                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                                16
.
For n = 7 we have that 4*7-1 = 27 and the 27th row of A237593 is [14, 5, 3, 2, 1, 2, 2, 1, 2, 3, 5, 14] and the 26th row of A237593 is [14, 5, 2, 2, 2, 1, 1, 2, 2, 2, 5, 14] therefore between both Dyck paths there are four regions (or parts) of sizes [14, 6, 6, 14], so row 7 is [14, 6, 6, 14].
The sum of divisors of 27 is 1 + 3 + 9 + 27 = A000203(27) = 40. On the other hand the sum of the parts of the symmetric representation of sigma(27) is 14 + 6 + 6 + 14 = 40, equaling the sum of divisors of 27.

Cf. A000203, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239053, A239660, A239931, A239932, A239934, A244050, A245092, A262626.

A237590 a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 18, 19, 21, 23, 26, 27, 29, 30, 32, 33, 37, 39, 41, 42, 45, 47, 51, 52, 54, 55, 57, 58, 62, 64, 67, 68, 70, 72, 76, 77, 79, 80, 82, 84, 87, 89, 91, 92, 95, 98, 102, 104, 106, 107, 111, 112, 116, 118, 120, 121, 123, 125, 130, 131, 135, 136, 138, 140, 144, 147, 149, 150, 152, 154

Offset: 1

Omar E. Pol, Mar 31 2014

nonn

The total area (or total number of cells) of the diagram after n stages is equal to A024916(n), the sum of all divisors of all positive integers <= n.
Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(n), see example.
For more information see A237593 and A237270.
a(n) is also the total number of terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Apr 20 2016

			Illustration of initial terms:
.                                                         _ _ _ _
.                                           _ _ _        |_ _ _  |_
.                               _ _ _      |_ _ _|       |_ _ _|   |_
.                     _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.             _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.       _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.      |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.
.       1      2        4          5            7              8
.
For n = 6 the diagram contains 8 regions (or parts), so a(6) = 8.
The sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 33. On the other hand after 6 stages the sum of all parts of the diagram is [1] + [3] + [2+2] + [7] + [3+3] + [12] = 33, equaling the sum of all divisors of all positive integers <= 6.
Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(6) = 3.
From _Omar E. Pol_, Dec 25 2020: (Start)
Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
|_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
(End)

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

Partial sums of A237271.
Compare with A060831 (analog for the diagram that contains subparts).
Cf. A000203, A004125, A024916, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239659, A239660, A239663, A239665, A239931-A239934, A245092, A244050, A244970, A262626, A317109.

(* total number of parts in the first n symmetric representations *)
(* Function a237270[] is defined in A237270 *)
(* variable "previous" represents the sum from 1 through m-1 *)
a237590[previous_,{m_,n_}]:=Rest[FoldList[Plus[#1,Length[a237270[#2]]]&,previous,Range[m,n]]]
a237590[n_]:=a237590[0,{1,n}]
a237590[78] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)

a(n) = A317109(n) - A294723(n) + 1 (Euler's formula). - Omar E. Pol, Jul 21 2018

Definition clarified by Omar E. Pol, Jul 21 2018

A112610 Number of representations of n as a sum of two squares and two triangular numbers.

Original entry on oeis.org

1, 6, 13, 14, 18, 32, 31, 30, 48, 38, 42, 78, 57, 54, 80, 62, 84, 96, 74, 96, 121, 108, 90, 128, 98, 102, 192, 110, 114, 182, 133, 156, 176, 160, 138, 192, 180, 150, 234, 158, 192, 288, 183, 174, 240, 182, 228, 320, 194, 198, 272, 252, 240, 288, 256, 252, 403, 230

Offset: 0

James Sellers, Dec 21 2005

nonn

Also row sums of A239931, hence the sequence has a symmetric representation. - Omar E. Pol, Aug 30 2015

			a(1) = 6 since we can write 1 = 1^2 + 0^2 + 0 + 0 = (-1)^2 + 0^2 + 0 + 0 = 0^2 + 1^2 + 0 + 0 = 0^2 + (-1)^2 + 0 + 0 = 0^2 + 0^2 + 1 + 0 = 0^2 + 0^2 + 0 + 1

Amiram Eldar, Table of n, a(n) for n = 0..10001
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

Cf. A000203, A193553, A237593, A239052, A239053, A239931.

[DivisorSigma(1, 4*n+1): n in [0..60]]; // Vincenzo Librandi, Sep 18 2015

Table[DivisorSigma[1, 4 n + 1], {n, 0, 57}] (* Michael De Vlieger, Aug 31 2015 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^14/eta(x+A)^6/eta(x^4+A)^4, n))} /* Michael Somos, Jul 04 2006 */

a(n) = sigma(4n+1) where sigma(n) = A000203(n) is the sum of the divisors of n.
Euler transform of period 4 sequence [ 6, -8, 6, -4, ...]. - Michael Somos, Jul 04 2006
Expansion of q^(-1/4)eta^14(q^2)/(eta^6(q)eta^4(q^4)) in powers of q. - Michael Somos, Jul 04 2006
Expansion of psi(q)^2*phi(q)^2, i.e., convolution of A004018 and A008441 [Hirschhorn]. - R. J. Mathar, Mar 24 2011
Sum_{k=0..n} a(k) = (Pi^2/4) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

A239665 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma of the smallest number whose symmetric representation of sigma has n parts.

Original entry on oeis.org

1, 2, 2, 5, 3, 5, 11, 5, 5, 11, 32, 12, 16, 12, 32, 74, 26, 14, 14, 26, 74, 179, 61, 29, 38, 29, 61, 179, 452, 152, 68, 32, 32, 68, 152, 452, 1250, 418, 182, 152, 100, 152, 182, 418, 1250, 3035, 1013, 437, 342, 85, 85, 342, 437, 1013, 3035, 6958, 1394, 638, 314, 154, 236, 154, 314, 638, 1394, 6958

Offset: 1

Omar E. Pol, Mar 23 2014

Row n is also row A239663(n) of A237270.

			----------------------------------------------------------------------
n    A239663(n)  Triangle begins:                        A266094(n)
----------------------------------------------------------------------
1        1       [1]                                         1
2        3       [2, 2]                                      4
3        9       [5, 3, 5]                                  13
4       21       [11, 5, 5, 11]                             32
5       63       [32, 12, 16, 12, 32]                      104
6      147       [74, 26, 14, 14, 26, 74]                  228
7      357       [179, 61, 29, 38, 29, 61, 179]            576
8      903       [452, 152, 68, 32, 32, 68, 152, 452]     1408
...
Illustration of initial terms:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.     _ _ 2          | |
.    |_ _|_ 2        | |
.     _ 1| |         | |
.    |_| |_|         |_|
.
For n = 2 we have that A239663(2) = 3 is the smallest number whose symmetric representation of sigma has 2 parts. Row 3 of A237593 is [2, 1, 1, 2] and row 2 of A237593 is [2, 2] therefore between both Dyck paths in the first quadrant there are two regions (or parts) of sizes [2, 2], so row 2 is [2, 2].
For n = 3 we have that A239663(3) = 9 is the smallest number whose symmetric representation of sigma has 3 parts. The 9th row of A237593 is [5, 2, 2, 2, 2, 5] and the 8th row of A237593 is [5, 2, 1, 1, 2, 5] therefore between both Dyck paths in the first quadrant there are three regions (or parts) of sizes [5, 3, 5], so row 3 is [5, 3, 5].

Cf. A000203, A005279, A196020, A236104, A237270, A237271, A235791, A237591, A237593, A239660, A239663, A239931-A239934, A240020, A240062, A244050, A245092, A262626, A266094.

a(16)-a(28) from Michel Marcus and Omar E. Pol, Mar 28 2014
a(29)-a(36) from Michel Marcus, Mar 28 2014
a(37)-a(45) from Michel Marcus, Mar 29 2014
a(46)-a(66) from Michel Marcus, Apr 02 2014

A319529 Odd numbers that have middle divisors.

Original entry on oeis.org

1, 9, 15, 25, 35, 45, 49, 63, 77, 81, 91, 99, 117, 121, 135, 143, 153, 165, 169, 187, 195, 209, 221, 225, 231, 247, 255, 273, 285, 289, 299, 315, 323, 325, 345, 357, 361, 375, 391, 399, 405, 425, 435, 437, 441, 459, 475, 483, 493, 513, 525, 527, 529, 551, 561, 567, 575, 589, 609, 621, 625, 627, 651

Offset: 1

Omar E. Pol, Sep 23 2018

nonn

Odd numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
From Felix Fröhlich, Sep 25 2018: (Start)
For the definition of middle divisors, see A067742.
Let t be a term of A005408. Then t is in this sequence iff A067742(t) != 0. (End)
From Hartmut F. W. Hoft, May 24 2022: (Start)
By Theorem 1 (iii) in A067742, the number of middle divisors of a(n) equals the width of the symmetric representation of sigma(a(n)) on the diagonal which equals the triangle entry A249223(n, A003056(n)).
All terms in sequence A016754 have an odd number of middle divisors, forming a subsequence of this sequence; A016754(18) = a(116) = 1225 = 5^2 * 7^2 is the smallest number in A016754 with 3 middle divisors: 25, 35, 49.
Sequence A259417 is a subsequence of this sequence and of A320137 since an even power of a prime has a single middle divisor.
The maximum widths of the center part of the symmetric representation of sigma(a(n)), SRS(a(n)), need not occur at the diagonal. For example, a(304) = 3^3 * 5^3 = 3375, SRS(3375) has 3 parts, its center part has maximum width 3 while its width at the diagonal equals 2 = A067742(3375), and divisors 45 and 75 are the two middle divisors of a(304). (End)

			9 is in the sequence because it's an odd number and the symmetric representation of sigma(9) = 13 has an odd number of parts (more exactly three parts), as shown below:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.                    | |
.                    | |
.                    | |
.                    |_|
.

Intersection of A005408 and A071562.
For more information about the diagram see A237593.
Cf. A000203, A067742, A071090, A236104, A237270, A237271, A239931, A239933, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319796, A319801, A319802.
Cf. A016754, A237048, A249223, A259417, A320137.

middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#Hartmut F. W. Hoft, May 24 2022 *)

from itertools import islice, count
from sympy import divisors
def A319529_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(1,startvalue+1-(startvalue&1)),2):
        if any((k <= 2*d**2 < 4*k for d in divisors(k,generator=True))):
            yield k
A319529_list = list(islice(A319529_gen(startvalue=11),40)) # Chai Wah Wu, Jun 09 2022

A241558 Smallest part of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 3, 2, 7, 3, 12, 4, 15, 3, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 5, 18, 12, 60, 5, 21, 6, 56, 15, 72, 16, 63, 7, 27, 12, 91, 19, 30, 8, 90, 21, 96, 22, 42, 23, 36, 24, 124, 7, 15, 10, 49, 27, 120, 8, 120, 11, 45, 30, 168, 31, 48, 12, 127, 9, 144, 34, 63, 13

Offset: 1

Michel Marcus and Omar E. Pol, Apr 29 2014

nonn

If A237271(n) = 1 then a(n) = A241559(n) = A241838(n) = A000203(n).
If n is an odd prime then a(n) = (n + 1)/2 = A241559(n) = A241838(n).
For more information see A237270 and A237593.

			For n = 9 the symmetric representation of sigma(9) = 13 in the first quadrant looks like this:
y
.
._ _ _ _ _ 5
|_ _ _ _ _|
.         |_ _ 3
.         |_  |
.           |_|_ _ 5
.               | |
.               | |
.               | |
.               | |
. . . . . . . . |_| . . x
.
There are three parts [5, 3, 5] and the smallest part is 3 so a(9) = 3.
For n = 45 the symmetric representation of sigma(45) = 78 has three parts [23, 32, 23] and the smallest part is 23 so a(45) = 23.
For n = 63 the symmetric representation of sigma(63) = 104 has five parts [32, 12, 16, 12, 32] and the smallest part is 12 so a(63) = 12.

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A000203, A071561, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A241559, A241838, A245092, A262626.

(* Function a237270[] is defined in A237270 *)
a241558[n_]:=Min[a237270[n]]
Map[a241558,Range[64]] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)

More terms from Jinyuan Wang, Feb 14 2020

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

A340833 a(n) is the number of vertices in the diagram of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 6, 7, 10, 9, 12, 11, 14, 14, 15, 13, 18, 13, 17, 20, 22, 15, 22, 15, 22, 23, 21, 17, 26, 22, 21, 25, 28, 19, 30, 19, 30, 27, 23, 26, 32, 21, 25, 29, 34, 21, 34, 21, 33, 36, 27, 23, 38, 30, 38, 31, 35, 23, 38, 35, 42, 33, 29, 25, 42, 25, 29, 42, 42, 37, 44, 27

Offset: 1

Omar E. Pol, Jan 23 2021

If A237271(n) is odd then a(n) is even.
If A237271(n) is even then a(n) is odd.
The above sentences arise that the diagram is always symmetric for any value of n hence the number of edges is always an even number. Also from Euler's formula.
Indices of odd terms give A071561.
Indices of even terms give A071562.
For another version with subparts see A340847 from which first differs at a(6).
The parity of this sequence is also the characteristic function of numbers that have no middle divisors (cf. A348327). - Omar E. Pol, Oct 14 2021

			Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             |   |_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      6        7         10           9              12
.
For n = 6 the diagram has 12 vertices so a(6) = 12.
On the other hand the diagram has 12 edges and only one part or region, so applying Euler's formula we have that a(6) = 12 - 1 + 1 = 12.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
n:              7                    8                      9
a(n):          11                   14                     14
.
For n = 9 the diagram has 14 vertices so a(9) = 14.
On the other hand the diagram has 16 edges and three parts or regions, so applying Euler's formula we have that a(9) = 16 - 3 + 1 = 14.
Another way for the illustration of initial terms is as follows:
--------------------------------------------------------------------------
.  n  a(n)                             Diagram
--------------------------------------------------------------------------
            _
   1   4   |_|  _
              _| |  _
   2   6     |_ _| | |  _
                _ _|_| | |  _
   3   7       |_ _|  _| | | |  _
                  _ _|  _| | | | |  _
   4  10         |_ _ _|  _|_| | | | |  _
                    _ _ _|  _ _| | | | | |  _
   5   9           |_ _ _| |    _| | | | | | |  _
                      _ _ _|  _|  _|_| | | | | | |  _
   6  12             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  11               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  14                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  14                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|    _ _| | | | | | | | |
  10  15                     |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |      _| |  _ _ _| | | | | |
  11  13                       |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  18                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |    _| |  _ _ _|
  13  13                           |_ _ _ _ _ _ _| | |  _|  _|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  17                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  20                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  22                                 |_ _ _ _ _ _ _ _ _|
...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n) for n=1..10^4, accentuating a(m) for m=1..2^8 for clarity, and labeling a(k) for k=1..24.

Parity gives A348327.
Cf. A237271 (number of parts or regions).
Cf. A340846 (number of edges).
Cf. A340847 (number of vertices in the diagram with subparts).
Cf. A294723 (total number of vertices in the unified diagram).
Cf. A239931-A239934 (illustration of first 32 diagrams).
Cf. A000203, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A237590, A237591, A237593, A239660, A245092, A262626, A340848.

MapAt[# + 1 &, #, 1] &@ Map[Length@ Union[Join @@ #] - 1 &, Partition[Prepend[#, {{0, 0}}], 2, 1]] &@ Table[{{0, 0}}~Join~Accumulate[Join[#, Reverse[Reverse /@ (-1*#)]]] &@ MapIndexed[Which[#2 == 1, {#1, 0}, Mod[#2, 2] == 0, {0, #1}, True, {-#1, 0}] & @@ {#1, First[#2]} &, If[Length[#] == 0, {n, n}, Join[{n}, #, {n - Total[#]}]]] &@ Differences[n - Array[(Ceiling[(n + 1)/# - (# + 1)/2]) &, Floor[(Sqrt[8 n + 1] - 1)/2]]], {n, 67}] (* Michael De Vlieger, Oct 27 2021 *)

a(n) = A340846(n) - A237271(n) + 1 (Euler's formula).

Terms a(33) and beyond from Michael De Vlieger, Oct 27 2021

cp's OEIS Frontend

A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.

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A239932 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-2).

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A239933 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-1).

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A237590 a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.

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A112610 Number of representations of n as a sum of two squares and two triangular numbers.

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Magma

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A239665 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma of the smallest number whose symmetric representation of sigma has n parts.

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A319529 Odd numbers that have middle divisors.

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A241558 Smallest part of the symmetric representation of sigma(n).

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