A253258 -id:A253258 - cp's OEIS frontend

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

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

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.

A376829 Numbers m whose symmetric representation of sigma(m) has at least a part with maximum width 3.

Original entry on oeis.org

60, 72, 84, 90, 126, 140, 144, 168, 198, 210, 216, 264, 270, 280, 288, 300, 312, 315, 330, 390, 396, 400, 440, 450, 462, 468, 495, 510, 520, 525, 528, 546, 560, 570, 576, 585, 588, 612, 616, 624, 648, 675, 684, 693, 702, 714, 728, 765, 770, 798, 800, 810, 816, 819, 828, 880, 882

Offset: 1

Hartmut F. W. Hoft, Oct 05 2024

nonn

All terms m in this sequence for which SRS(m) consists of 1 or 2 parts are even.
Let m = 2^k * q, k >= 0 and q > 2 odd, be a number in this sequence. Let c be the number of divisors r <= A003056(m) of q for which there is at most one pair of divisors s and t of q satisfying r < s < t <= min( 2^(k+1) * r, A003056(m)). Call such triples (r, s, t) good triples. Then at least one good triple exists for number m.
Let w be the number of times that width 3 occurs in the width pattern of m (row m in the triangle of A341969). Then c = (w + 1)/2 when the width at the diagonal is equal to 3 and c = w/2 otherwise.

			a(1) = 60 has one good triple 1 < 3 < 5 of odd divisors which determines 2 width 3 occurrences in its width pattern 1 2 3 2 3 2 1, and SRS(60) has width 2 at the diagonal.
a(2) = 72 has one good triple 1 < 3 < 9 of odd divisors which determines 1 width 3 occurrence in its unimodal width pattern 1 2 3 2 1, and SRS(72) has width 3 at the diagonal.
a(18) = 315 is the smallest odd number in the sequence and SRS(315) has three parts. SRS(a(1)) .. SRS(a(17)) each consists of a single part.
a(41) = 648 = 2^3 * 3^4 has two good triples 1 < 3 < 9 and 3 < 9 < 27 of odd divisors which determine 3 width 3 occurrences in its width pattern 1 2 3 2 3 2 3 2 1, and SRS(648) has width 3 at the diagonal.
a(57) = 882 has two good triples  1 < 7 < 9 and 7 < 9 < 21 of odd divisors which determine 4 width 3 occurrences in its width pattern is 1 2 1 2 3 2 3 2 1 2 3 2 3 2 1 2 1, and SRS(882) has width 1 at the diagonal.
a(514) = 7620 is the smallest number with 2 parts in its symmetric representation of sigma. It has two good triples  1 < 3 < 5 and 3 < 5 < 15 of odd divisors which determine 4 width 3 occurrences in its width pattern 1 2 3 2 3 2 1 0 1 2 3 2 3 2 1 and width 0 at the diagonal.
a(734) = 10728 is the smallest number in the sequence for which SRS(10728) has 2 parts and 2 occurrences of width 3. Each of its 2 parts therefore is unimodal: 1 2 3 2 1 0 1 2 3 2 1.

Column 3 of A253258.

Cf. A003056, A237591, A237593, A249223, A341969.

(* t249223[n] is row n in A249223, widthPattern[ ] is defined in A341969 *)
t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]]
widthPattern[n_] := Map[First, Split[Join[t249223[n], Reverse[t249223[n]]]]]
a376829[m_, n_] := Select[Range[m, n], Max[widthPattern[#]]==3&]
a376829[1, 900]

A347273 Number of positive widths in the symmetric representation of sigma(n).

Original entry on oeis.org

1, 3, 4, 7, 6, 11, 8, 15, 13, 18, 12, 23, 14, 24, 23, 31, 18, 35, 20, 39, 32, 36, 24, 47, 31, 42, 40, 55, 30, 59, 32, 63, 48, 54, 45, 71, 38, 60, 56, 79, 42, 83, 44, 84, 73, 72, 48, 95, 57, 93, 72, 98, 54, 107, 72, 111

Offset: 1

Omar E. Pol, Aug 29 2021

a(n) is also the number of columns that contain ON cells in the ziggurat diagram of n. Both diagrams can be unified in a three-dimensional version.
a(n) is also the number of nonzero terms in the n-th row of A249351.
The number of widths in the symmetric representation of sigma(n) is equal to 2*n - 1 = A005408(n-1).
The sum of the positive widths (also the sum of all widths) of the symmetric representation of sigma(n) equals A000203(n).
Indices where a(n) = 2*n - 1 give A174973 and also A238443.
a(p) = p + 1, if p is prime.
a(n) = 2*n - 1, if and only if A237271(n) = 1.
a(n) = A000203(n) if n is a member of A174905.
For the definition of "width" see A249351.

Cf. A000040, A005408, A174905, A174973, A196020, A235791, A236106, A237270, A237271, A237591, A237593, A238443, A249351 (widths), A253258, A347361.

a(n) = A005408(n-1) - A347361(n).

A347361 Number of widths that are zero in the symmetric representation of sigma(n).

Original entry on oeis.org

0, 0, 1, 0, 3, 0, 5, 0, 4, 1, 9, 0, 11, 3, 6, 0, 15, 0, 17, 0, 9, 7, 21, 0, 18, 9, 13, 0, 27, 0, 29, 0, 17, 13, 24, 0, 35, 15, 21, 0, 39, 0, 41, 3, 16, 19, 45, 0, 40, 6, 29, 5, 51, 0, 37, 0

Offset: 1

Omar E. Pol, Aug 29 2021

a(n) is also the number of columns without ON square cells in the ziggurat diagram of n. Both diagrams can be unified in a three-dimensional version.
a(n) is also the number of zeros in the n-th row of A249351.
The number of widths in the symmetric representation of sigma(n) is equal to 2*n - 1 = A005408(n-1).
The sum of the widths of the symmetric representation of sigma(n) equals A000203(n).
a(n) = 0, if and only if A237271(n) = 1.
a(p) = p - 2, if p is prime.
For the definition of "width" see A249351.

Indices of zeros give A174973 and also A238443.

Cf. A000040, A005408, A196020, A235791, A236106, A237270, A237271, A237591, A237593, A249351 (widths), A253258, A347273.

a(n) = A005408(n-1) - A347273(n).

A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 63, 66, 70, 75, 77, 78, 80, 88, 91, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 130, 132, 135, 138, 143, 150, 153, 154, 156, 160, 162, 165, 170, 174, 175, 176, 182, 186, 187, 189, 190, 192, 195, 196, 200

Offset: 1

Hartmut F. W. Hoft, Aug 21 2024

nonn

Number m = 2^k * q, k >= 0 and q odd, is in this sequence precisely when for any divisor s <= A003056(m) of q there is at most one divisor t of q satisfying s < t <= min(2^(k+1) * s, A003056(m)), and at least one such pair s < t of successive odd divisors exists. Equivalently, row m of the triangle in A249223 contains at least one 2, but no number larger than 2.

			a(4) = 18 has width pattern 1 2 1 2 1 in its symmetric representation of sigma consisting of a single part, and row 18 in the triangle of A249223 is 1 1 2 1 1.
a(9) = 35 has width pattern 1 0 1 2 1 0 1 in its symmetric representation of sigma consisting of 3 parts, and row 35 in the triangle of A249223 is 1 0 0 0 1 1 2.
Irregular triangle of rows a(n) in triangle of A341970, i.e. of positions of 1's in triangle of A237048, and for the corresponding widths to the diagonal in triangle of A341969:
a(n)| row in A341970      left half of row in A341969
6   | 1   3               1   2
12  | 1   3               1   2
15  | 1   2   3   5       1   0   1   2
18  | 1   3   4           1   2   1
20  | 1   5               1   2
24  | 1   3               1   2
28  | 1   7               1   2
30  | 1   3   4   5       1   2   1   2
35  | 1   2   5   7       1   0   1   2
36  | 1   3   8           1   2   1
...

Column 2 of A253258.
Subsequence of A005279.
Some subsequences are A352030, A370205, A370206, A370209.
Cf. A003056, A174905, A235791, A237593, A249223, A249351, A250068, A341969, A341970.

eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
mW2Q[n_] := Max[FoldWhileList[#1+If[OddQ[#2], 1, -1]&, sDiv[n], #1<=2&]]==2
a375611[m_, n_] := Select[Range[m, n], mW2Q]
a375611[1, 200]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

A376829 Numbers m whose symmetric representation of sigma(m) has at least a part with maximum width 3.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A347273 Number of positive widths in the symmetric representation of sigma(n).

Views

Author

Keywords

Comments

Crossrefs

Formula

A347361 Number of widths that are zero in the symmetric representation of sigma(n).

Views

Author

Keywords

Comments

Crossrefs

Formula

A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica