A241008 -id:A241008 - cp's OEIS frontend

A249351 Triangle read by rows in which row n lists the widths of the symmetric representation of sigma(n).

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Oct 26 2014

Here T(n,k) is defined to be the "k-th width" of the symmetric representation of sigma(n), with n>=1 and 1<=k<=2n-1. Explanation: consider the diagram of the symmetric representation of sigma(n) described in A236104, A237593 and other related sequences. Imagine that the diagram for sigma(n) contains 2n-1 equidistant segments which are parallel to the main diagonal [(0,0),(n,n)] of the quadrant. The segments are located on the diagonal of the cells. The distance between two parallel segment is equal to sqrt(2)/2. T(n,k) is the length of the k-th segment divided by sqrt(2). Note that the triangle contains nonnegative terms because for some n the value of some widths is equal to zero. For an illustration of some widths see Hartmut F. W. Hoft's contribution in the Links section of A237270.
Row n has length 2*n-1.
Row sums give A000203.
If n is a power of 2 then all terms of row n are 1's.
If n is an even perfect number then all terms of row n are 1's except the middle term which is 2.
If n is an odd prime then row n lists (n+1)/2 1's, n-2 zeros, (n+1)/2 1's.
The number of blocks of positive terms in row n gives A237271(n).
The sum of the k-th block of positive terms in row n gives A237270(n,k).
It appears that the middle diagonal is also A067742 (which was conjectured by Michel Marcus in the entry A237593 and checked with two Mathematica functions up to n = 100000 by Hartmut F. W. Hoft).
It appears that the trapezoidal numbers (A165513) are also the numbers k > 1 with the property that some of the noncentral widths of the symmetric representation of sigma(k) are not equal to 1. - Omar E. Pol, Mar 04 2023

			Triangle begins:
  1;
  1,1,1;
  1,1,0,1,1;
  1,1,1,1,1,1,1;
  1,1,1,0,0,0,1,1,1;
  1,1,1,1,1,2,1,1,1,1,1;
  1,1,1,1,0,0,0,0,0,1,1,1,1;
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
  1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,1,1;
  1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1;
  1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1;
  1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1;
  ...
---------------------------------------------------------------------------
.        Written as an isosceles triangle              Diagram of
.              the sequence begins:               the symmetry of sigma
---------------------------------------------------------------------------
.                                                _ _ _ _ _ _ _ _ _ _ _ _
.                      1;                       |_| | | | | | | | | | | |
.                    1,1,1;                     |_ _|_| | | | | | | | | |
.                  1,1,0,1,1;                   |_ _|  _|_| | | | | | | |
.                1,1,1,1,1,1,1;                 |_ _ _|    _|_| | | | | |
.              1,1,1,0,0,0,1,1,1;               |_ _ _|  _|  _ _|_| | | |
.            1,1,1,1,1,2,1,1,1,1,1;             |_ _ _ _|  _| |  _ _|_| |
.          1,1,1,1,0,0,0,0,0,1,1,1,1;           |_ _ _ _| |_ _|_|    _ _|
.        1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;         |_ _ _ _ _|  _|     |
.      1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,1,1;       |_ _ _ _ _| |      _|
.    1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1;     |_ _ _ _ _ _|  _ _|
.  1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1;   |_ _ _ _ _ _| |
.1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1; |_ _ _ _ _ _ _|
...
From _Omar E. Pol_, Nov 22 2020: (Start)
Also consider the infinite double-staircases diagram defined in A335616.
For n = 15 the diagram with first 15 levels looks like this:
.
Level                         "Double-staircases" diagram
.                                          _
1                                        _|1|_
2                                      _|1 _ 1|_
3                                    _|1  |1|  1|_
4                                  _|1   _| |_   1|_
5                                _|1    |1 _ 1|    1|_
6                              _|1     _| |1| |_     1|_
7                            _|1      |1  | |  1|      1|_
8                          _|1       _|  _| |_  |_       1|_
9                        _|1        |1  |1 _ 1|  1|        1|_
10                     _|1         _|   | |1| |   |_         1|_
11                   _|1          |1   _| | | |_   1|          1|_
12                 _|1           _|   |1  | |  1|   |_           1|_
13               _|1            |1    |  _| |_  |    1|            1|_
14             _|1             _|    _| |1 _ 1| |_    |_             1|_
15            |1              |1    |1  | |1| |  1|    1|              1|
.
Starting from A196020 and after the algorithm described in A280850 and A296508 applied to the above diagram we have a new diagram as shown below:
.
Level                             "Ziggurat" diagram
.                                          _
6                                         |1|
7                            _            | |            _
8                          _|1|          _| |_          |1|_
9                        _|1  |         |1   1|         |  1|_
10                     _|1    |         |     |         |    1|_
11                   _|1      |        _|     |_        |      1|_
12                 _|1        |       |1       1|       |        1|_
13               _|1          |       |         |       |          1|_
14             _|1            |      _|    _    |_      |            1|_
15            |1              |     |1    |1|    1|     |              1|
.
The 15th row
of this seq:  [1,1,1,1,1,1,1,1,0,0,0,1,1,1,2,1,1,1,0,0,0,1,1,1,1,1,1,1,1]
The 15th row
of A237270:   [              8,            8,            8              ]
The 15th row
of A296508:   [              8,      7,    1,    0,      8              ]
The 15th row
of A280851    [              8,      7,    1,            8              ]
.
The number of horizontal steps (or 1's) in the successive columns of the above diagram gives the 15th row of this triangle.
For more information about the parts of the symmetric representation of sigma(n) see A237270. For more information about the subparts see A239387, A296508, A280851.
More generally, it appears there is the same correspondence between the original diagram of the symmetric representation of sigma(n) and the "Ziggurat" diagram of n. (End)

Index entries for sequences related to sigma(n)

Cf. A000203, A003056, A067742, A071562, A165513, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A238443, A239660, A239932-A239934, A240542, A241008, A241010, A245092, A245685, A246955, A246956, A247687, A249223, A250068, A250070, A250071, A262626, A280850, A280851, A296508, A235616, A347186.

(* function segments are defined in A237270 *)
a249351[n_] := Flatten[Map[segments, Range[n]]]
a249351[10] (* Hartmut F. W. Hoft, Jul 20 2022 *)

A071561 Numbers with no middle divisors (cf. A071090).

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 111, 113, 114

Offset: 1

Robert G. Wilson v, May 30 2002

nonn

Numbers k such that A071090(k) is 0.
Conjecture: lim_{n->oo} a(n)/n = 4/3.
Regarding the above conjecture, numerical calculations suggest that this limit is smaller than 4/3. See A071540. - Amiram Eldar, Jul 27 2024
Also numbers n with the property that the number of parts in the symmetric representation of sigma(n) is even. - Michel Marcus and Omar E. Pol, Apr 25 2014 [For a proof see the link. - Hartmut F. W. Hoft, Sep 09 2015]
Middle divisors are divisors d with sqrt(k/2) <= d < sqrt(2k). - Michael B. Porter, Oct 19 2018

			From _Michael B. Porter_, Oct 19 2018: (Start)
The divisors of 21 are 1, 3, 7, and 21.  Since none of these are between sqrt(21/2) = 3.24... and sqrt(2*21) = 6.48..., 21 is in the sequence.
The divisors of 20 are 1, 2, 4, 5, 10, and 20.  Since 4 and 5 are both between sqrt(20/2) = 3.16... and sqrt(2*20) = 6.32..., 20 is not in the sequence. (End)

Iain Fox, Table of n, a(n) for n = 1..10000
Hartmut F. W. Hoft, On the symmetric spectrum of odd divisors of a number, (2015), (Note that in this paper, A241561 should be replaced with A071561, and A241562 should be replaced with A071562. Also note that "the symmetric spectrum of odd divisors of a number" seems to be an attempt to call with a new name to a diagram known since 2014 as "the symmetric representation of sigma(n)"). - _Omar E. Pol_, Oct 08 2018
José Manuel Rodríguez Caballero, Elementary number-theoretical statements proved by Language Theory, arXiv:1709.09617 [math.LO], 2017.
J. M. Rodríguez Caballero, Symmetric Dyck Paths and Hooley's Δ-Function, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.

Cf. A071090, A071540, A071562, A071563, A237048, A237270, A237271, A237593, A241008.

f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[125], f[ # ] == 0 &]
(* Related to the symmetric representation of sigma *)
(* subsequence of even parts of number k for m <= k <= n *)
(* Function a237270[] is defined in A237270 *)
(* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *)
a071561[m_, n_]:=Select[Range[m, n], EvenQ[Length[a237270[#]]]&]
a071561[1, 114] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)
Select[Range@ 120, Function[n, Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] == {}]] (* Michael De Vlieger, Jan 03 2017 *)

is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1 \\ Iain Fox, Dec 19 2017

is(n,f=factor(n))=my(t=(n+1)\2); fordiv(f,d, if(d^2>=t, return(d^2>2*n))); 0 \\ Charles R Greathouse IV, Jan 22 2018

list(lim)=my(v=List(),t); forfactored(n=3,lim\1, t=(n[1]+1)\2; fordiv(n[2],d, if(d^2>=t, if(d^2>2*n[1], listput(v,n[1])); break))); Vec(v) \\ Charles R Greathouse IV, Jan 22 2018

A250068 Maximum width of any region in the symmetric representation of sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2

Offset: 1

Hartmut F. W. Hoft, Nov 11 2014

nonn

Since the width of the single region of the symmetric representation of sigma( 2^ceiling((p-1)*(log_2 3) - 1) * 3^(p-1) ), for prime number p, at the diagonal equals p, this sequence contains an increasing subsequence (see A250071).

a(n) is also the number of layers of width 1 in the symmetric representation of sigma(n). For more information see A001227. - Omar E. Pol, Dec 13 2016

			a(6) = 2 since the sequence of widths at each unit step in the symmetric representation of sigma(6) = 12 is 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1. For visual examples see A237270, A237593 and sequences referenced in these.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A000203, A001227, A237048, A237270, A237271, A237591, A237593, A241008, A241010, A246955, A247687, A249223, A249351 (widths), A279387, A279388, A279391.

See A250070, A250071, A340506 for records.

(* function a2[ ] is defined in A249223 *)
a250068[n_]:=Max[a2[n]]
a250068[{m_,n_}]:=Map[a250068,Range[m,n]]
a250068[{1,100}](* data *)

t237048(n,k) = if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0);
kmax(n) = (sqrt(1+8*n)-1)/2;
t249223(n,k) = sum(j=1, k, (-1)^(j+1)*t237048(n,j));
a(n) = my(wm = t249223(n, 1)); for (k=2, kmax(n), wm = max(wm, t249223(n, k))); wm; \\ Michel Marcus, Sep 20 2015

a(n) = max_{k=1..floor((sqrt(8*n+1) - 1)/2)} (Sum_{j=1..k}(-1)^(j+1)*A237048(n, j)), for n >= 1.

A250070 Smallest number k such that the symmetric representation of sigma(k) has at least one part of width n.

Original entry on oeis.org

1, 6, 60, 120, 360, 840, 3360, 2520, 5040, 10080, 15120, 32760, 27720, 50400, 98280, 83160, 110880, 138600, 221760, 277200, 332640, 360360, 554400, 960960, 831600, 942480, 720720, 2217600, 1965600, 1441440, 3160080, 2827440, 2162160, 2882880, 3603600, 5765760, 5654880, 4324320, 9979200

Offset: 1

Hartmut F. W. Hoft, Nov 11 2014

nonn

The 26 entries starting with a(2) = 6 are products of powers of consecutive primes starting with 2, except for a(12) = 32760 and a(15) = 98280 (which are missing 11), and a(26) = 942480 (which is missing 13).
a(n) is the smallest number k such that the symmetric representation of sigma(k) has n layers. For more information see A279387. - Omar E. Pol, Dec 16 2016
Row 1 of A253258. - Omar E. Pol, Apr 15 2018
From Hartmut F. W. Hoft, Jun 10 2024: (Start)
All terms a(n) <= 1.75*10^7 have a symmetric representation of sigma that consists of a single part and they are abundant for n > 2. Numbers a(1) = 1, a(2) = 6, and a(4) = 120 are unimodal while numbers a(6) = 840, a(14) = 50400, a(18) = 138600, a(24) = 960960, a(26) = 942480, a(32) = 2827440, a(44) = 8648640 have a single extent of maximum width, but are not unimodal.
Conjecture: The symmetric representation of sigma for every term consists of a single part and it is unimodal only for a(1), a(2), and a(4).
As a consequence, this sequence would be a subsequence of A174973, and all a(n), n > 2, would be abundant. (End)

			a(3) = 60 since the symmetric representation of sigma(60) = 168 consists of a single region of whose successive widths are 41 1's, 9 2's, 6 3's, 7 2's, 6 3's, 9 2's, and 41 1's.
a(6) = 840 has a single extent of 12 units of width 6 centered around point (583,583) on the diagonal, but is not unimodal. - _Hartmut F. W. Hoft_, Jun 10 2024

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..48

Indices of records in A250068.
Cf. A000203, A237270, A237271, A237593, A241008, A241010, A246955, A247687, A249223, A250071, A253258, A279387.
Cf. A174973.

(* function a2[ ] is defined in A249223 *)
a250070[{j_, k_}, b_] := Module[{i, max, acc={{1, 1}}}, For[i=j, i<=k, i++, max={Max[a2[i]], i}; If[max[[1]]>b && !MemberQ[Transpose[acc][[1]], max[[1]]], AppendTo[acc,max]]]; acc]
(* returns (argument,result) data pairs since sequence is non-monotonic *)
Sort[a250070[{1, 1000000}, 1]] (* computed in steps *)

a(n) = min(k such that A250068(k) = n), n >= 1.

a(28)-a(48) from Hartmut F. W. Hoft, Jun 10 2024

A246955 Numbers j for which the symmetric representation of sigma(j) has two parts, each of width one.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 92, 94, 97, 101, 103, 106, 107, 109, 113, 116, 118, 122, 124, 127, 131, 134, 136, 137, 139, 142, 146, 148, 149, 151, 152, 157, 158, 163, 164, 166, 167, 172, 173, 178, 179, 181, 184, 188, 191, 193, 194, 197, 199

Offset: 1

Hartmut F. W. Hoft, Sep 08 2014

nonn

The sequence is the intersection of A239929 (sigma(j) has two parts) and of A241008 (sigma(j) has an even number of parts of width one).
The numbers in the sequence are precisely those defined by the formula for the triangle, see the link. The symmetric representation of sigma(j) has two parts, each part having width one, precisely when j = 2^(k - 1) * p where 2^k <= row(j) < p, p is prime and row(j) = floor((sqrt(8*j + 1) - 1)/2). Therefore, the sequence can be written naturally as a triangle as shown in the Example section.
The symmetric representation of sigma(j) = 2*j - 2 consists of two regions of width 1 that meet on the diagonal precisely when j = 2^(2^m - 1)*(2^(2^m) + 1) where 2^(2^m) + 1 is a Fermat prime (see A019434). This subsequence of numbers j is 3, 10, 136, 32896, 2147516416, ...[?]... (A191363).
The k-th column of the triangle starts in the row whose initial entry is the first prime larger than 2^(k+1) (that sequence of primes is A014210, except for 2).
Observation: at least the first 82 terms coincide with the numbers j with no middle divisors whose largest divisor <= sqrt(j) is a power of 2, or in other words, coincide with the intersection of A071561 and A365406. - Omar E. Pol, Oct 11 2023

			We show portions of the first eight columns, 0 <= k <= 7, of the triangle.
0    1    2     3     4     5     6     7
3
5    10
7    14
11   22   44
13   26   52
17   34   68    136
19   38   76    152
23   46   92    184
29   58   116   232
31   62   124   248
37   74   148   296   592
41   82   164   328   656
43   86   172   344   688
47   94   188   376   752
53   106  212   424   848
59   118  236   472   944
61   122  244   488   976
67   134  268   536   1072  2144
71   142  284   568   1136  2272
.    .    .     .     .     .
.    .    .     .     .     .
127  254  508   1016  2032  4064
131  262  524   1048  2096  4192  8384
137  274  548   1096  2192  4384  8768
.    .    .     .     .     .     .
.    .    .     .     .     .     .
251  502  1004  2008  4016  8032  16064
257  514  1028  2056  4112  8224  16448  32896
263  526  1052  2104  4208  8416  16832  33664
Since 2^(2^4) + 1 = 65537 is the 6543rd prime, column k = 15 starts with 2^15*(2^(2^16) + 1) = 2147516416 in row 6542 with 65537 in column k = 0.
For an image of the symmetric representations of sigma(m) for all values m <= 137 in the triangle see the link.
The first column is the sequence of odd primes, see A065091.
The second column is the sequence of twice the primes starting with 10, see A001747.
The third column is the sequence of four times the primes starting with 44, see A001749.
For related references also see A033676 (largest divisor of n less than or equal to sqrt(n)).

Hartmut F. W. Hoft, Equivalence proof of sequence and triangle
Hartmut F. W. Hoft, Image for sigma(n) values

Cf. A000203, A033676, A071561, A163280, A237270, A237271, A237593, A241008, A241010, A247687, A250068, A250070, A250071, A365406.

(* functions path[] and a237270[ ] are defined in A237270 *)
atmostOneDiagonalsQ[n_]:=SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], - 1] - path[n - 1], 1]]]
(* data *)
Select[Range[200], Length[a237270[#]]==2 && atmostOneDiagonalsQ[#]&]
(* function for computing triangle in the Example section through row 55 *)
TableForm[Table[2^k Prime[n], {n, 2, 56}, {k, 0, Floor[Log[2, Prime[n]] - 1]}], TableDepth->2]

Formula for the triangle of numbers associated with the sequence:

P(n, k) = 2^k * prime(n) where n >= 2, 0 <= k <= floor(log_2(prime(n)) - 1).

A174905 Numbers with no pair (d,e) of divisors such that d < e < 2*d.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 76, 79, 81, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 98, 101, 103, 106

Offset: 1

Reinhard Zumkeller, Apr 01 2010

nonn

A174903(a(n)) = 0; complement of A005279;
sequences of powers of primes are subsequences;
a(n) = A129511(n) for n < 27, A129511(27) = 35 whereas a(27) = 37.
Also the union of A241008 and A241010 (see the link for a proof). - Hartmut F. W. Hoft, Jul 02 2015
In other words: numbers n with the property that all parts in the symmetric representation of sigma(n) have width 1. - Omar E. Pol, Dec 08 2016
Sequence A357581 shows the numbers organized in columns of a square array by the number of parts in their symmetric representation of sigma. - Hartmut F. W. Hoft, Oct 04 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Hartmut F. W. Hoft, Proof that this sequence equals union of A241008 and A241010

Cf. A000040, A000961, A001248, A005279, A030078, A030514, A129511, A174903, A237271, A237593, A241008, A241010.

Cf. A357581.

a174905 n = a174905_list !! (n-1)
a174905_list = filter ((== 0) . a174903) [1..]
-- Reinhard Zumkeller, Sep 29 2014

filter:= proc(n)
  local d,q;
   d:= numtheory:-divisors(n);
   min(seq(d[i+1]/d[i],i=1..nops(d)-1)) >= 2
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 08 2014

(* it suffices to test adjacent divisors *)
a174905[n_] := Module[{d = Divisors[n]}, ! Apply[Or, Map[2 #[[1]] > #[[2]] &, Transpose[{Drop[d, -1], Drop[d, 1]}]]]]
(* Hartmut F. W. Hoft, Aug 07 2014 *)
Select[Range[106], !MatchQ[Divisors[#], {_, d_, e_, _} /; e < 2d]& ] (* Jean-François Alcover, Jan 31 2018 *)

A247687 Numbers m with the property that the symmetric representation of sigma(m) has three parts of width one.

Original entry on oeis.org

9, 25, 49, 50, 98, 121, 169, 242, 289, 338, 361, 484, 529, 578, 676, 722, 841, 961, 1058, 1156, 1369, 1444, 1681, 1682, 1849, 1922, 2116, 2209, 2312, 2738, 2809, 2888, 3362, 3364, 3481, 3698, 3721, 3844, 4232, 4418, 4489, 5041, 5329, 5476, 5618, 6241, 6724, 6728, 6889, 6962, 7396, 7442, 7688, 7921, 8836, 8978, 9409

Offset: 1

Hartmut F. W. Hoft, Sep 22 2014

nonn

The symmetric representation of sigma(m) has 3 regions of width 1 where the two extremal regions each have 2^k - 1 legs and the central region starts with the p-th leg of the associated Dyck path for sigma(m) precisely when m = 2^(k - 1) * p^2 where 2^k < p <= row(m), k >= 1, p >= 3 is prime and row(m) = floor((sqrt(8*m + 1) - 1)/2). Furthermore, the areas of the two outer regions are (2^k - 1)*(p^2 + 1)/2 each so that the area of the central region is (2^k - 1)*p; for a proof see the link.
Since the sequence is defined by a two-parameter expression it can be written naturally as a triangle as shown in the Example section.
A263951 is a subsequence of this sequence containing the squares of all those primes p for which the areas of the 3 regions in the symmetric representation of p^2 (p once and (p^2 + 1)/2 twice) are primes; i.e., p^2 and p^2 + 1 are semiprimes (see A070552). - Hartmut F. W. Hoft, Aug 06 2020

			We show portions of the first eight columns, powers of two 0 <= k <= 7, and 55 rows of the triangle through prime(56) = 263.
p/k     0       1       2       3       4       5       6       7
3       9
5       25      50
7       49      98
11      121     242     484
13      169     338     676
17      289     578     1156    2312
19      361     722     1444    2888
23      529     1058    2116    4232
29      841     1682    3364    6728
31      961     1922    3844    7688
37      1369    2738    5476    10952   21904
41      1681    3362    6724    13448   26896
43      1849    3698    7396    14792   29584
47      2209    4418    8836    17672   35344
53      2809    5618    11236   22472   44944
59      3481    6962    13924   27848   55696
61      3721    7442    14884   29768   59536
67      4489    8978    17956   35912   71824   143648
71      5041    10082   20164   40328   80656   161312
.       .       .       .       .       .       .
.       .       .       .       .       .       .
131     17161   34322   68644   137288  274567  549152  1098304
137     18769   37538   75076   150152  300304  600608  1201216
.       .       .       .       .       .       .       .
.       .       .       .       .       .       .       .
257     66049   132098  264196  528392  1056784 2113568 4227136 8454272
263     69169   138338  276676  553352  1106704 2213408 4426816 8853632
Number 4 is not in this sequence since the symmetric representation of sigma(4) consists of a single region. Column k=0 contains the squares of primes (A001248(n), n>=2), column k=1 contains double the squares of primes (A079704(n), n>=2) and column k=2 contains four times the squares of primes (A069262(n), n>=5).

Hartmut F. W. Hoft, Three regions width one - triangle formula proof

Cf. A000203, A237270, A237271, A237593, A241008, A241010, A246955, A250068, A250070, A250071.

Cf. A070552, A263951.

(* path[n] and a237270[n] are defined in A237270 *)
atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]
(* data *)
Select[Range[10000], atmostOneDiagonalsQ[#] && Length[a237270[#]]==3 &]
(* expression for the triangle in the Example section *)
TableForm[Table[2^k Prime[n]^2, {n, 2, 57}, {k, 0, Floor[Log[2, Prime[n]] - 1]}], TableDepth -> 2, TableHeadings -> {Map[Prime, Range[2, 57]], Range[0, Floor[Log[2, Prime[57] - 1]]]}]

As an irregular triangle, T(n, k) = 2^k * prime(n)^2 where n >= 2 and 0 <= k <= floor(log_2(prime(n)) - 1).

A250071 Smallest number k such that the symmetric representation of sigma(k) has maximum width n for those k whose representation has nondecreasing width up to the diagonal.

Original entry on oeis.org

1, 6, 72, 120, 5184, 1440, 373248, 6720, 28800, 103680, 1934917632, 80640, 278628139008, 7464960, 2073600, 483840, 1444408272617472, 1612800, 103997395628457984, 5806080, 298598400, 77396705280, 539122498937926189056, 7096320, 1658880000, 5572562780160, 90316800, 418037760, 402452788967166148425547776, 116121600

Offset: 1

Hartmut F. W. Hoft, Nov 11 2014

nonn

The symmetric representation of sigma(k) has nondecreasing width to the diagonal precisely when all odd divisors counted in the k-th row of A237048 occur at odd indices. If we write k = 2^m * q with m >= 0 and q odd, this property is equivalent to q < 2^(m+1).
The values for a(11), a(13), a(17) and a(19) were computed directly using the formula k = 2^m * 3^(p-1) where p is one of the four primes and m the smallest exponent so that 3^(p-1) < 2^(m+1). Each of these numbers has a symmetric representation of nondecreasing width ending in a prime number width, and they are the first such numbers since the number of divisors of an odd number is a prime precisely when the number is a power of a prime.
The other numbers listed whose symmetric representations of sigma(k) have nondecreasing width are smaller than 7500000. The only additional numbers k <= 100000000 are a(24) = 7096320, a(27) = 90316800 and a(32) = 85155840.
See A340506 for another way to look at this data. - N. J. A. Sloane, Jan 23 2021

			a(6) = 1440 = 2^5 * 3^2 * 5 has 6 odd divisors. It is the smallest number of the form 2^m * q with m > 0, q odd and such that q < 2^(m+1).

Cf. A000203, A237048, A237270, A237271, A237593, A241008, A241010, A246955, A247687, A249223, A250068, A250070.
Cf. also A340506.
Cf. A162247.

(* function a2[ ] is defined in A249223 *)
smallQ[n_] := Module[{x=2^IntegerExponent[n,2]}, n/x<2x]
ndWidth[{m_,n_}] := Select[Range[m, n], smallQ]
a250071[x_List] := Module[{i, max, acc={{1, 1}}}, For[i=1, i<=Length[x], i++, max={Max[a2[x[[i]]]], x[[i]]}; If[!MemberQ[Transpose[acc][[1]], max[[1]]], AppendTo[acc, max]]]; acc]
(* returns (argument,result) data pairs since sequence is non-monotonic *)
Sort[a250071[ndWidth[{1,100000000}]]] (* computed in steps *)
(* alternate implementation using function f[ ] by T. D. Noe in A162247 *)
sF[n_] := Min[Map[Apply[Times, Prime[Range[2, Length[#]+1]]^#]&, Map[Reverse[#-1]&, f[n]]]]
f1U[n_] := Module[{s=sF[n], k}, k=Floor[Log[2, s]]; 2^k s]
a250071[n_] := Map[f1U, Range[n]]
a250071[30] (* Hartmut F. W. Hoft, Nov 27 2024 *)

a(n) = min(2^m * q, m >= 0 & q odd & sigma_0(q) = n & q < 2^(m+1)) where sigma_0 is the number of divisors.

a(p) = 2^ceiling((p-1)*(log_2(3)) - 1) * 3^(p-1) for primes p.

a(21)-a(30) from Hartmut F. W. Hoft, Nov 27 2024

A241010 Numbers n with the property that the number of parts in the symmetric representation of sigma(n) is odd, and that all parts have width 1.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 25, 32, 49, 50, 64, 81, 98, 121, 128, 169, 242, 256, 289, 338, 361, 484, 512, 529, 578, 625, 676, 722, 729, 841, 961, 1024, 1058, 1156, 1250, 1369, 1444, 1681, 1682, 1849, 1922, 2048, 2116, 2209, 2312, 2401, 2738, 2809, 2888, 3025, 3249, 3362, 3364, 3481, 3698, 3721, 3844

Offset: 1

Hartmut F. W. Hoft, Aug 07 2014

nonn

The first eight entries in A071562 but not in this sequence are 6, 12, 15, 18, 20, 24, 28 & 30.
The first eight entries in A238443 but not in this sequence are 6, 12, 18, 20, 24, 28, 30 & 36.
The union of A241008 and of this sequence equals A174905 (for a proof see link in A174905).
Let n = 2^m * product(p_i^e_i, i=1,...,k) = 2^m * q with m >= 0, k >= 0, 2 < p_1, ...< p_k primes and e_i >= 1, for all 1 <= i <= k. For each number n in this sequence all e_i are even, and for any two odd divisors f < g of n, 2^(m+1) * f < g. The sum of the areas of the regions r(n, z) equals sigma(n). For a proof of the characterization and the formula see the theorem in the link below.
Numbers 3025 = 5^2 * 11^2 and 510050 = 2^1 * 5^2 * 101^2 are the smallest odd and even numbers, respectively, in the sequence with two distinct odd prime factors.
Among the 706 numbers in the sequence less than 1000000 (see link to the table) there are 143 that have two different odd prime factors, but none with three. All numbers with three different odd prime factors are larger than 500000000. Number 4450891225 = 5^2 * 11^2 * 1213^2 is in the sequence, but may not be the smallest one with three different odd prime factors. Note that 1213 is the first prime that extends the list of divisors of 3025 while preserving the property for numbers in this sequence.
The subsequence of numbers n = 2^(k-1) * p^2 satisfying the constraints above is A247687.
n = 3^(2*h) = 9^h = A001019(h), h>=0, is the smallest number for which the symmetric representation of sigma(n) has 2*h+1 regions of width one, for example for h = 1, 2, 3 and 5, but not for h = 4 in which case 3025 = 5^2 * 11^2 < 3^8 = 6561 is the smallest (see A318843). [Comment corrected by Hartmut F. W. Hoft, Sep 04 2018]
Computations using this characterization are more efficient than those of Dyck paths for the symmetric representations of sigma(n), e.g., the Mathematica code below.

			This irregular triangle presents in each column those elements of the sequence that have the same factor of a power of 2.
  row/col      2^0    2^1   2^2   2^3    2^4    2^5  ...
   2^k:          1      2     4     8     16     32  ...
   3^2:          9
   5^2:         25     50
   7^2:         49     98
   3^4:         81
  11^2:        121    242   484
  13^2:        169    338   676
  17^2:        289    578  1156  2312
  19^2:        361    722  1444  2888
  23^2:        529   1058  2116  4232
   5^4:        625   1250
   3^6:        729
  29^2:        841   1682  3364  6728
  31^2:        961   1922  3844  7688
  37^2:       1369   2738  5476 10952 21904
  41^2:       1681   3362  6724 13448 26896
  43^2:       1849   3698  7396 14792 29584
  47^2:       2209   4418  8836 17672 35344
   7^4:       2401   4802
  53^2:       2809   5618 11236 22472 44944
  5^2*11^2:   3025
  3^2*19^2:   3249
  59^2:       3481   6962 13924 27848 55696
  61^2:       3721   7442 14884 29768 59536
  67^2:       4489   8978 17956 35912 71824 143648
  3^2*23^2:   4761
  71^2:       5041
  ...
  5^2*101^2:225025 510050
  ...
Number 3025 = 5^2 * 11^2 is in the sequence since its divisors are 1, 5, 11, 25, 55, 121, 275, 605 and 3025. Number 6050 = 2^1 * 5^2 * 11^2 is not in the sequence since 2^2 * 5 > 11 while 5 < 11.
Number 510050 = 2^1 * 5^2 * 101^2 is in the sequence since its 9 odd divisors 1, 5, 25, 101, 505, 2525, 10201, 51005 and 225025 are separated by factors larger than 2^2. The areas of its 9 regions are 382539, 76515, 15339, 3939, 1515, 3939, 15339, 76515 and 382539. However, 2^2 * 5^2 * 101^2 is not in the sequence.
The first row is A000079.
The rows, except the first, are indexed by products of even powers of the odd primes satisfying the property, sorted in increasing order.
The first column is a subsequence of A244579.
A row labeled p^(2*h), h>=1 and p>=3 with p = A000040(n), has A098388(n) entries.
Starting with the second column, dividing the entries of a column by 2 creates a proper subsequence of the prior column.
See A259417 for references to other sequences of even powers of odd primes that are subsequences of column 1.
The first entry greater than 16 in column labeled 2^4 is 21904 since 37 is the first prime larger than 2^5. The rightmost entry in the row labeled 19^2 is 2888 in the column labeled 2^3 since 2^4 < 19 < 2^5.

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..563 (values less than 1000000)
Hartmut F. W. Hoft, Proof of characterization theorem
Hartmut F. W. Hoft, Table of n, a(n) for n = 1...706(values less than 1000000)
Hartmut F. W. Hoft, Illustration of the symmetric representation of sigma(a(n)), for n=1..15
Hartmut F. W. Hoft, Proof of property for n and formula for regions of sigma(n)

Cf. A000203, A174905, A236104, A237270 (symmetric representation of sigma(n)), A237271, A237593, A238443, A241008, A071562, A246955, A247687, A250068, A250070, A250071.

Cf. A318843

(* path[n] and a237270[n] are defined in A237270 *)
atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]
Select[Range[1000], atmostOneDiagonalsQ[#] && OddQ[Length[a237270[#]]]&] (* data *)
(* more efficient code based on numeric characterization *)
divisorPairsQ[m_, q_] := Module[{d = Divisors[q]}, Select[2^(m + 1)*Most[d] - Rest[d], # >= 0 &] == {}]
a241010AltQ[n_] := Module[{m, q, p, e}, m=IntegerExponent[n, 2]; q=n/2^m; {p, e} = Transpose[FactorInteger[q]]; q==1||(Select[e, EvenQ]==e && divisorPairsQ[m, q])]
a241010Alt[m_,n_] := Select[Range[m, n], a241010AltQ]
a241010Alt[1,4000] (* data *)

Formula for the z-th region in the symmetric representation of n = 2^m * q in this sequence, 1 <= z <= sigma_0(q) and q odd: r(n, z) = 1/2 * (2^(m+1) - 1) * (d_z + d_(2*x+2-z)) where 1 = d_1 < ... < d_(2*x+1) = q are the odd divisors of n.

More terms and further edited by Hartmut F. W. Hoft, Jun 26 2015 and Jul 02 2015 and corrected Oct 11 2015

A262611 Triangle read by rows in which row n lists the widths of the symmetric representation of A024916(n): the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 3, 2, 1, 1, 2, 3, 3, 3, 3, 3, 2, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 4, 5, 4, 4, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 5, 6, 5, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 5, 6, 7, 6, 5, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 6, 7, 7, 7, 6, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 6, 6, 7, 7, 7, 6, 6, 6, 6, 5, 4, 3, 2, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Here T(n,k) is defined to be the "k-th width" of the symmetric representation of A024916(n), with n>=1 and 1<=k<=2n-1.
If both A249351 and this sequence are written as isosceles triangles then the partial sums of the columns of A249351 give the columns of this isosceles triangle (see the second triangle in Example section).
For the definition of the k-th width of the symmetric representation of sigma(n) see A249351.
Note that for the geometric representation of the n-th row of the triangle we need the x-axis, the y-axis, and only a Dyck path which is given by the elements of the n-th row of the triangle A237593.
Row n has length 2*n-1.
Row sums give A024916.
The middle diagonal is A240542.

			Triangle begins:
1;
1,2,1;
1,2,2,2,1;
1,2,3,3,3,2,1;
1,2,3,3,3,3,3,2,1;
1,2,3,4,4,5,4,4,3,2,1;
1,2,3,4,4,4,5,4,4,4,3,2,1;
1,2,3,4,5,5,5,6,5,5,5,4,3,2,1;
1,2,3,4,5,5,5,6,7,6,5,5,5,4,3,2,1;
1,2,3,4,5,6,6,6,7,7,7,6,6,6,5,4,3,2,1;
1,2,3,4,5,6,6,6,6,7,7,7,6,6,6,6,5,4,3,2,1;
1,2,3,4,5,6,7,7,7,8,9,9,9,8,7,7,7,6,5,4,3,2,1;
...
--------------------------------------------------------------------------
.        Written as an isosceles triangle
.              the sequence begins:               Diagram for n = 1..12
--------------------------------------------------------------------------
.                                                _ _ _ _ _ _ _ _ _ _ _ _
.                      1;                       |_| | | | | | | | | | | |
.                    1,2,1;                     |_ _|_| | | | | | | | | |
.                  1,2,2,2,1;                   |_ _|  _|_| | | | | | | |
.                1,2,3,3,3,2,1;                 |_ _ _|    _|_| | | | | |
.              1,2,3,3,3,3,3,2,1;               |_ _ _|  _|  _ _|_| | | |
.            1,2,3,4,4,5,4,4,3,2,1;             |_ _ _ _|  _| |  _ _|_| |
.          1,2,3,4,4,4,5,4,4,4,3,2,1;           |_ _ _ _| |_ _|_|    _ _|
.        1,2,3,4,5,5,5,6,5,5,5,4,3,2,1;         |_ _ _ _ _|  _|     |
.      1,2,3,4,5,5,5,6,7,6,5,5,5,4,3,2,1;       |_ _ _ _ _| |      _|
.    1,2,3,4,5,6,6,6,7,7,7,6,6,6,5,4,3,2,1;     |_ _ _ _ _ _|  _ _|
.  1,2,3,4,5,6,6,6,6,7,7,7,6,6,6,6,5,4,3,2,1;   |_ _ _ _ _ _| |
.1,2,3,4,5,6,7,7,7,8,9,9,9,8,7,7,7,6,5,4,3,2,1; |_ _ _ _ _ _ _|
...
For n = 3 the symmetric representation of A024916(3) = 8 in the 4th quadrant looks like this:
.
.    Polygon         Cells
.     _ _ _          _ _ _
.    |     |        |_|_|_|
.    |    _|        |_|_|_|
.    |_ _|          |_|_|
.
There are eight cells. The representation of the widths looks like this:
.
.     \ \ \
.     \ \ \
.     \ \    1
.          2 2
.        1 2
.
So the third row of the triangle is [1, 2, 2, 2, 1].

Cf. A024916, A236104, A237048, A237593, A240542, A241008, A241010, A244250, A245685, A246955, A247687, A249351, A250068, A250070, A250071, A253258, A262626.

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A249351 Triangle read by rows in which row n lists the widths of the symmetric representation of sigma(n).

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A071561 Numbers with no middle divisors (cf. A071090).

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A250068 Maximum width of any region in the symmetric representation of sigma(n).

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A250070 Smallest number k such that the symmetric representation of sigma(k) has at least one part of width n.

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A246955 Numbers j for which the symmetric representation of sigma(j) has two parts, each of width one.

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A174905 Numbers with no pair (d,e) of divisors such that d < e < 2*d.

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A247687 Numbers m with the property that the symmetric representation of sigma(m) has three parts of width one.

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