A342592 -id:A342592 - cp's OEIS frontend

A342594 Earliest occurrence of the next distinct width pattern (as listed in A342592) in the symmetric representation of sigma(n) not yet encountered as n increases.

1, 3, 6, 9, 15, 18, 21, 30, 45, 60, 63, 72, 75, 78, 81, 90, 105, 120, 135, 147, 150, 162, 165, 180, 189, 210, 225, 231, 300, 315, 357, 360, 378, 390, 405, 420, 441, 450, 465, 495, 504, 525, 540, 567, 630, 648, 666, 675, 690, 693, 729, 735, 770, 810, 825, 840, 855, 858, 882, 900, 903, 945, 975, 990

Offset: 1

Hartmut F. W. Hoft, Mar 16 2021

nonn

The width pattern of the symmetric representation of sigma(a(n)) is the n-th row of the table of A342592.

Conjecture: If for some number n the symmetric representation of sigma(n) has the symmetric width pattern w in row n of A342592 then infinitely many numbers have that width pattern w.

			a(1) = 1 is the smallest power of 2 whose symmetric representation of sigma has width pattern (1).
a(2) = 3 is the smallest odd prime whose symmetric representation of sigma has width pattern (1 0 1).
a(4) = 9 is the first number whose symmetric representation of sigma has width pattern (1 0 1 0 1). The infinitely many numbers 2^s * p^2, s >= 0 and p an odd prime larger than 2^(s+1), have the same width pattern.

Cf. A235791, A237048, A237270, A237591, A237593, A249223, A341969, A342592.

(* function a341969[ ] is defined in A341969 *)
a342594[n_] := Module[{listW={}, listK={}, k, w}, For[k=1, k<=n, k++, w=a341969[k]; If[!MemberQ[listW, w], AppendTo[listW, w]; AppendTo[listK, k]]]; listK]
a342594[990] (* 64 entries; the 64th new pattern is encountered at n=990 *)

A342595 Irregular triangle of A342592 read by rows arranged first by length of the width pattern and then lexicographically within blocks of patterns of equal length.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 1, 2, 3, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 2, 1, 0, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 2, 3, 2, 1

Offset: 1

Hartmut F. W. Hoft, Mar 16 2021

Let u and v be two symmetric width patterns; then u < v if u is shorter than v, and if they have the same length then they are ordered lexicographically, i.e., if i is the first index where u and v differ and u(i) < v(i) then u < v.
This sequence is a permutation of the rows of the irregular triangle in A342592. Every row in the triangle representing a width pattern w contains an odd number 2*k - 1, k >= 1, of entries where k is the number of odd divisors of the smallest number whose symmetric representation of sigma realizes that pattern.
The number of distinct width patterns w of the same length 2*k-1 created by numbers with k odd divisors is computationally challenging since the numbers of their first occurrence can be very large (A342592, A342596). The counts in the table below are already established for n <= 5*10^5 and have not changed through 10^7; the counts are not stable at that larger level for width patterns of numbers with more than 8 odd divisors:
# odd divisors   1   2   3   4    5    6    7    8
pattern count    1   2   3   6    5   16    7   40
A001405          1   2   3   6   10   20   35   70
For any odd number q with k divisors and 2^s < q < 2^(s+1), s >= 0, any number 2^t * q with t > s has the lexicographically largest symmetric width pattern 1 2 3 ... k-2 k-1 k k-1 k-2 ... 3 2 1 of length 2*k - 1. Therefore, the sequence q, 2 * q, 2^2 * q, ... , 2^s * q instantiates at most s+1 different symmetric width patterns; these range from 2 for prime numbers q, patterns (1 0 1) and (1 2 1), to the maximum of s+1 different patterns such as for q = 105 = 3*5*7.

			The number of entries through the center in a row of the triangle below equals the number of odd divisors of any number that has that pattern of widths.
The pattern in row 10 of the triangle below, realized first by n = 30 which labels the row is the smallest number with width pattern (1 2 1 2 1 2 1); 42, 54 and 66 are the other numbers less than 100 realizing that pattern.
The triangle below lists the first 21 distinct symmetric width patterns in the order described above. The smallest number whose symmetric representation of sigma has the width pattern of that row is listed as first column (see A342596). All possible symmetric width patterns of lengths 1, 3, 5 and 7 are realized in the triangle below; their respective counts are A001405(1,2,3,4) = (1,2,3,6).
     1   1
     3   1 0 1
     6   1 2 1
     9   1 0 1 0 1
    18   1 2 1 2 1
    72   1 2 3 2 1
    21   1 0 1 0 1 0 1
    15   1 0 1 2 1 0 1
    78   1 2 1 0 1 2 1
    30   1 2 1 2 1 2 1
    60   1 2 3 2 3 2 1
   120   1 2 3 4 3 2 1
    81   1 0 1 0 1 0 1 0 1
   162   1 2 1 2 1 2 1 2 1
   648   1 2 3 2 3 2 3 2 1
  1296   1 2 3 4 3 4 3 2 1
  5184   1 2 3 4 5 4 3 2 1
   147   1 0 1 0 1 0 1 0 1 0 1
    63   1 0 1 0 1 2 1 0 1 0 1
    75   1 0 1 2 1 0 1 2 1 0 1
    45   1 0 1 2 1 2 1 2 1 0 1

Cf. A001405, A237593, A249223, A341969, A342592, A342594, A342596.

(* function a341969[ ] is defined in A341969 *)
lexicographic[s1_, s2_] := Module[{k=1}, While[s1[[2, k]]==s2[[2, k]], k++]; s1[[2,k]]

A342596 Numbers k of the earliest occurrence of widths patterns in the symmetric representation of sigma listed in the ordering of patterns in A342595.

Original entry on oeis.org

1, 3, 6, 9, 18, 72, 21, 15, 78, 30, 60, 120, 81, 162, 648, 1296, 5184, 147, 63, 75, 45, 1014, 666, 150, 90, 10728, 3816, 300, 180, 27744, 504, 360, 1440, 729, 1458, 5832, 11664, 46656, 93312, 373248, 903, 357, 189, 231, 465, 165, 105, 135, 1001, 770, 12246, 4134, 1482, 1326, 1830, 690, 390, 858, 210, 378

Offset: 1

Hartmut F. W. Hoft, Mar 16 2021

nonn

This sequence is the companion to A342595 in that a(n) is the smallest number k that has row n of the table in A342595 as its width pattern in the symmetric representation of sigma(k).

The number of possible width patterns of length n occurring up to the diagonal in symmetric representations of sigma is A001405(n). Those are realized for n <= 4. For larger n the actual number of width patterns is smaller. Only p symmetric patterns of length 2p-1 are realizable when a number has p odd divisors and p is prime. Patterns such as 1 0 1 2 3 ... k-1 k k-1 ... 3 2 1 0 1, k >= 4, i.e., numbers with at least 6 odd divisors, cannot be realized as width patterns in the symmetric representation of sigma. If n = 2^s * p * q^2, s >= 0, p < q odd primes, then 2^(s+1) < p and row(n) < 2^(s+1) * p must hold which leads to the contradiction q^2 < p^2; if n = 2^s * p^2 * q, s >= 0, p < q odd primes, then again 2^(s+1) < p and row(n) < 2^(s+1) * p must hold which leads to the contradiction p * q < p^2.

			a(17) = 5184 = 2^6 * 3^4 is the smallest number with width pattern (1 2 3 4 5 4 3 2 1).
a(18) = 147 = 3 * 7^2 is the smallest number with width pattern (1 0 1 0 1 0 1 0 1 0 1).

Cf. A001405, A237593, A249223, A341969, A342592, A342594, A342595.

(* a341969[] defined in A341969 and lexicographicOrder[] in A342595 *)
a342596[n_] := Module[{listW={}, listK={}, k, w}, For[k=1, k<=n, k++, w=a341969[k]; If[!MemberQ[listW, w], AppendTo[listW, w]; AppendTo[listK, k]]]; Flatten[Map[First, Sort[Transpose[{listK, listW}], lexicographicOrder]]]]
Take[a342596[500000], 60]

A367377 Square array T(n, k), n >= 1, k >= 1, read by antidiagonals, of the least numbers whose symmetric representation of sigma instantiate the unimodal width pattern 1, 2, ..., n, ..., 2, 1 repeated k times separated by instances of width 0.

Original entry on oeis.org

1, 6, 3, 72, 78, 9, 120, 10728, 1014, 21, 5184, 28920, 1598472, 12246, 81, 1440, 53752896, 6969720, 230297976, 171366, 147, 373248, 4157280

Offset: 1

Hartmut F. W. Hoft, Nov 15 2023

The numbers T(n, 1) instantiating a single unimodal pattern of width n form A250071(n). This first column is not increasing since T(5, 1) = 5184 > 1440 = T(6, 1).
The numbers T(1, k) instantiating the repeating unimodal patterns 1, 1, 0, 1, ..., 1, 0, 1, 0, ..., 0, 1, 0, 1, ... of width 1 form A318843(k). This first row is not increasing since T(1, 11) = 59049 > 29095 = T(1, 12).
The rows in the table are infinite since the numbers T(n, 1) * p^(k-1) >= T(n, k), with p the smallest prime greater than 2 * T(n, 1), instantiate the width pattern for T(n, k), though equality need not hold, as T(1, 4) = 21 = 3 * 7 < 1 * 3^3 = 27 demonstrates.
Conjecture 1: None of the rows and columns are increasing.
Conjecture 2: T(n, p) = T(n, 1) * A151800(2*T(n, 1))^(p-1) for n >= 1 and primes p.
Conjecture 3: T(p, q), p and q primes, is a record for its upper left hand rectangle in the table. Only one prime number index generally is not sufficient as the inequality 4157280 = T(6, 2) < 5 * 10^6 < T(5, 2) shows.

			The corner of the table begins:
  --------------------------------------------------------------------
     Pattern   |   once    twice  3 times   4 times  5 times  6 times
  --------------------------------------------------------------------
        1      |      1        3        9        21       81      147
       121     |      6       78     1014     12246   171366  1922622
      12321    |     72    10728  1598472 230297976
     1234321   |    120    28920  6969720
    123454321  |   5184 53752896
   12345654321 |   1440  4157280
  1234567654321| 373248
  ...
T(3, 4) must have 12 odd divisors and as least number must have 2^3 * 3^2 as a factor in order to create the initial width pattern 1 2 3 2 1 0. Therefore, since the next smallest prime larger than 16 * 9 is 149, T(3, 4) is 2^3 * 3^2 * 149^3 or 2^3 * 3^2 * 149 * p for suitable prime p which leads to p = 21467 < 22201 = 149^2.
All other numbers in the table were found by exhaustive computations.

Cf. A235791, A237048, A237270, A237591, A237593, A249223, A250071, A262045, A318843, A341969, A342592, A342594, A342595, A342596.

t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]] (* row n in triangle of A249223 *)
t262045[n_] := Join[t249223[n], Reverse[t249223[n]]] (* row n in triangle of A262045 *)
widthPattern[n_] := Map[First, Split[t262045[n]]]
umw[n_, k_] := Most[Flatten[Table[Join[Range[n], Range[n-1, 0, -1]], k]]]
a367377[{n_, k_}, b_] := NestWhile[#+1&, 1, #

A370206 Numbers j whose symmetric representation of sigma(j) consists of two copies of unimodal width pattern 121 separated by 0.

Original entry on oeis.org

78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 348, 354, 366, 372, 402, 426, 438, 444, 474, 492, 498, 516, 534, 564, 582, 606, 618, 636, 642, 654, 678, 708, 732, 762, 786, 804, 820, 822, 834, 852, 860, 876, 894, 906, 940, 942, 948, 978, 996, 1002, 1038, 1060, 1068, 1074

Offset: 1

Hartmut F. W. Hoft, Feb 11 2024

nonn

Each term has 4 odd divisors and has the form 2^k * p * q, k > 0, p and q prime,  2 < p < 2^(k+1) <  2^(k+1) * p < q. The inequalities ensure that the four 1's in row a(n) of triangle in A237048 are in positions 1, p, 2^(k+1), and 2^(k+1) * p <= floor( (sqrt(8*a(n)+1) - 1)/2 ) < q and establish width pattern 1210 in SRS(a(n)) up to the diagonal. Also since p < 2^(k+1), numbers of the form 2^k * p^3 force p^2 < 2^(k+1) * p which creates a width pattern of the form 1212121.
When a(n) satisfies q = 2^(k+1) * p + 1 it is the smallest number with prime factor p  whose two parts of SRS(a(n)) meet at the diagonal since in this case 2^(k+1) * p = floor( (sqrt(8*a(n)+1) - 1)/2 ). The first 4 numbers with p = 3 are  2* 3 * 13 = 78, 2^4 * 3 * 97 = 4656, 2^5 * 3 * 193 = 18528 and 2^7 * 3 * 769 = 295296. The smallest number with prime factor p = 47 has 355 digits.
Conjecture: The subsequence of numbers m whose two parts of SRS(m) meet at the diagonal is infinite.

			a(1) = 78 = 2 * 3 * 13 = A262259(3) and SRS(78) consists of 2 unimodal parts of width pattern 121 that meet at diagonal position (54, 54).
a(38) = 4 * 5 * 41 = 820 = A262259(6)  is the smallest number in the sequence divisible by 5 and the two parts of SRS(a(38)) meet at diagonal position (570, 570).

Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A262259, A341969, A342592, A342594, A342595, A342596, A367377, A370205.

(* function based on conditions for the odd divisors - fast computation *)
a370206Q[n_] := Module[{f=FactorInteger[n], d=Divisors[NestWhile[#/2&, n, EvenQ[#]&]]}, Length[f]==3&&f[[1, 1]]==2&&Length[d]==4&&f[[2, 1]]<2^(f[[1, 2]]+1)&&2^(f[[1, 2]]+1)*f[[2, 1]]A367377 - slow computation *)
a370206[m_, n_] :=  Select[Range[m, n], widthPattern[#]=={1, 2, 1, 0, 1, 2, 1}&]
a370206[1,1074]

A370205 Numbers j whose symmetric representation of sigma(j) consists of the single unimodal width pattern 121.

Original entry on oeis.org

6, 12, 20, 24, 28, 40, 48, 56, 80, 88, 96, 104, 112, 160, 176, 192, 208, 224, 272, 304, 320, 352, 368, 384, 416, 448, 464, 496, 544, 608, 640, 704, 736, 768, 832, 896, 928, 992, 1088, 1184, 1216, 1280, 1312, 1376, 1408, 1472, 1504, 1536, 1664, 1696, 1792, 1856, 1888, 1952, 1984

Offset: 1

Hartmut F. W. Hoft, Feb 11 2024

nonn

Every term has 2 odd divisors and has the form 2^k * p, k > 0, p prime and 2 < p < 2^(k+1), and therefore is a subsequence of A082662. The two 1's in row a(n) of the triangle of A237048 occur in positions 1 and p up to the diagonal since p <= floor( (sqrt(8*a(n) + 1) - 1)/2 ) < 2^(k+1) which represents the unimodal width pattern 121 in SRS(a(n)).

Numbers in this sequence divisible by 5 have the form 2^(k+2) * 5, k >= 0, the least being a(3) = 20.

Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A341969, A342592, A342594, A342595, A342596.

(* function based on conditions for the odd divisors - fast computation *)
a370205Q[n_] := Module[{p=NestWhile[#/2&, n, EvenQ[#]&]}, PrimeQ[p]&&p^2<2n]
a370205[m_, n_] := Select[Range[m, n], a370205Q]
a370205[1, 1984]
(* widthPattern[ ] and support functions are defined in A341969 - slow computation *)
a370205[m_, n_] := Select[Range[m, n], widthPattern[#]=={1, 2, 1}&]
a370205[1, 1984]

A370209 a(n) is the smallest number of the form 2^k * p * (2^(k+1) * p + 1) where 2 < p < 2^(k+1) is the n-th prime and 2^(k+1) * p + 1 is prime, or -1 if no such number exists.

Original entry on oeis.org

78, 820, 6328, 62128, 5539456, 155155972096, 739936, 69342976, 431056, 31494016, 44864128, 3525354496, 3788128

Offset: 2

Hartmut F. W. Hoft, Feb 11 2024

nonn

a(n) is the smallest number of the form described above whose symmetric representation of sigma, SRS(a(n)), consists of 2 parts that have a unimodal width pattern of type 121 and that meet at the diagonal. Since floor( (sqrt(8*a(n) + 1) - 1)/2 ) = 2^(k+1) * p, the central 0 width extent of SRS(a(n)) equals 0.

Conjecture: The sequence is infinite.

			a(2) = 78 = 2 * 3 * 13 = A262259(3) and SRS(78) consists of 2 unimodal parts 121 that meet at diagonal position (54, 54).
a(4) = 6328 = 8 * 7 * 113 = A262259(11) which demonstrates that  2^k < p < 2^(k+1) need not be true.
a(15) with k = 582 and p = 47, its second prime factor 2^(k+1) * p + 1 has 178 digits so that a(15) has 355 digits.
a(16) = 24129129742336 = 2^16 * 53 * 6946817.
Table of records of number of digits a(2) through a(500):
sequence index    2  3  4  5  6   7   15    76   419    438
number of digits  2  3  4  5  7  12  355  3854  5856  20049

Chai Wah Wu, Table of n, a(n) for n = 2..75 [Notice that a(15) has 355 digits, so the graph cuts off after 14 terms. - _N. J. A. Sloane_, Feb 18 2024]

Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A262259, A341969, A342592, A342594, A342595, A342596, A367377, A370205, A370206.

minExp[p_] := Module[{k=Floor[Log[2, p]]}, NestWhile[#+1&, k+1, !PrimeQ[2^# p+1]&]-1]/;PrimeQ[p]
a370209[p_] := Module[{k=minExp[p]}, 2^k p(2^(k+1)p+1)]/;PrimeQ[p]
Map[a370209[Prime[#]]&, Range[2, 14]] (* a(15) is too large to list *)

from itertools import count
from sympy import prime, isprime
def A370209(n):
    p = prime(n)
    return next((p<Chai Wah Wu, Feb 17 2024

a(n) = min( 2^k * p * (2^(k+1) * p + 1) : p = prime(n), 2 < p < 2^(k+1), 2^(k+1) * p + 1 is prime ), n>=2.

A372180 Square array read by antidiagonals upwards in which T(n,m) is the n-th number whose symmetric representation of sigma consists of m copies of unimodal pattern 121 (separated by 0's if m > 1).

Original entry on oeis.org

6, 12, 78, 20, 102, 1014, 24, 114, 1734, 12246, 28, 138, 2166, 12714, 171366, 40, 174, 3174, 13026, 501126, 1922622, 48, 186, 5046, 13182, 781926, 2057406, 28960854, 56, 222, 5766, 13494, 1679046, 2067546, 144825414, 300014754, 80, 246, 8214, 13962, 4243686, 2072382, 282275286, 300137214, 4174476774

Offset: 1

Hartmut F. W. Hoft, Apr 21 2024

Every number in this sequence is even since the symmetric representation of sigma for an odd number q starts 101. Each number in column m of T(n,m) has 2*m odd divisors.
Since u(m) = 2 * 3 * 13^(m-1), m>=1, has 2m odd divisors and 1 < 3 < 4 < 4*3 < 13 < 3*13 < 4*13 < 3*4*13 < 13^2 < ..., the symmetric representation of sigma for u(m) consists of m copies of unimodal pattern 121. Therefore, every column in the table T(n,m), m>=1, contains infinitely many entries. Number u(m) is the smallest entry in the m-th column when m is prime.
In general: If m>1 then T(n,m) = 2^k * q, k>=1, q odd, has at least 4 odd divisors which satisfy
d_(2i+2)  < 2^(k+1) * d_(2i+1) < 2^(k+1) * d_(2i+2) < d_(2i+3), i>=0,
with the odd divisors d_j of n in increasing order.

			a(1) = T(1,1) = 6, its symmetric representation of sigma, SRS(6), has unimodal pattern 121 and a single unit of width 2 at the diagonal.
a(3) = T(1,2) = 78, SRS(78) has unimodal pattern 1210121;
a(10) = T(1,4) = 12246, SRS(12246) has unimodal pattern 121012101210121;
both symmetric representations of sigma have width 0 at the diagonal where two parts meets.
Each number in the m-th column has 2m odd divisors. T(1,9) = 4174476774.
  -------------------------------------------------------------------------
   n\m  1    2     3     4       5         6          7          8
  -------------------------------------------------------------------------
   1|   6   78   1014  12246   171366   1922622    28960854  300014754 ...
   2|  12  102   1734  12714   501126   2057406   144825414  300137214 ...
   3|  20  114   2166  13026   781926   2067546   282275286  300235182 ...
   4|  24  138   3174  13182   1679046  2072382   888215334  300357642 ...
   5|  28  174   5046  13494   4243686  2081742  3568939926  300431118 ...
   6|  40  186   5766  13962   5541126  2091882     ...      300602562 ...
   7|  48  222   8214  14118   8487372  2097966              300651546 ...
   8|  56  246  10086  14898  11082252  2110134              300896466 ...
   9|  80  258  10092  15054  11244966  2112162              301165878 ...
  10|  88  282  11094  15366  16954566  2116218              301386306 ...
  ...

Row 1 gives A372181.
Column 1 gives A370205.
Column 2 gives A370206.
Cf. A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A341969, A341971, A342592, A342594, A342595, A342596, A367377, A370209.

divQ[k_, {d1_, d2_, d3_}] := d2<2^(k+1)d1&&2^(k+1)d2

T(n,1) = 2^k * p with odd prime p satisfying p < 2^(k+1), see A370205.

T(n,2) = 2^k * p * q, k > 0, p and q prime, 2 < p < 2^(k+1) < 2^(k+1) * p < q, see A370206.

A367370 a(k) is the number of different widths patterns in the symmetric representation of sigma for numbers having k odd divisors.

Original entry on oeis.org

1, 2, 3, 6, 5, 16, 7, 40

Offset: 1

Hartmut F. W. Hoft, Dec 05 2023

The width pattern (A341969) of the symmetric representation of sigma for a number with k >= 1 odd divisors has length 2*k - 1.
a(p) = p for any prime number p is realized by the m+1 numbers 3^(p-1), ..., 2^m * 3^(p-1) which contain m+1-p duplicates, where m = floor(log_2(3^(p-1))). Each width pattern first increases to a level 1 <= i <= p and then alternates between i and i-1 up to the diagonal of the symmetric representation of sigma resulting in p distinct patterns.
For some numbers n = 2^m * q, q odd and not prime, that are the least instantiations of a width pattern their odd parts q may not be the least instantiations of a width pattern, examples are 78, 1014, 12246 and 171366 with 4, 6, 8 and 10 odd divisors, respectively (see row 2 of the table in A367377).
Conjecture: a(9) = 28.
The least number instantiating the 28th width pattern, 12345654345654321, is n = 43356672, found in a search up to 5*10^9.
Table of width pattern counts of the symmetric representation of sigma and of all possible symmetric patterns:
# odd divisors  1   2   3   4   5    6    7    8    9   10   11    12
pattern count   1   2   3   6   5   16    7   40   28? >=47  11 >=223
A001405         1   2   3   6  10   20   35   70  126  252  462   924
The 4 symmetric patterns 10123232101, 10123432101, 12101010121 and 12123432121 cannot be instantiated as width patterns of numbers with 6 odd divisors.
30 of the 70 possible symmetric patterns of numbers n = 2^m * q, m>=0 and q odd, with 8 odd divisors cannot be instantiated as width patterns of the symmetric representation of sigma(n) since their sequence of widths contradicts the order of the odd divisors d_i of n and of the numbers 2^(m+1) * d_i and the positions of their corresponding 1's in the rows of the triangle of widths in A249223.

			In the irregular triangle below, row k lists the count and the first occurrences of successive instantiations of the distinct width patterns in the symmetric representation of sigma for numbers with k odd divisors.
# div |count|    first occurrence of distinct width patterns
      |     |    1    2    3     4     5     6      7 .. 11 .. 16 .. 40
-----------------------------------------------------------------------
1     |  1  |    1                                        .     .     .
2     |  2  |    3    6                                   .     .     .
3     |  3  |    9   18   72                              .     .     .
4     |  6  |   15   21   30    60    78   120            .     .     .
5     |  5  |   81  162  648  1296  5184                  .     .     .
6     | 16  |   45   63   75    90   147   150    180    ...  27744   .
7     |  7  |  729 1458 5832 11664 46656 93312 373248     .           .
8     | 40  |  105  135  165   189   210   231    357    ...       203808
9     | 28? |  225  441  450   882   900  1225   1800    ...
10    | >=47|  405  567  810  1134  1377  1539   1620    ...
11    | 11  |59049                 ...               1934917632
The complete sequence of first occurrences of the 11 width patterns for numbers with 11 odd divisors is: 59049, 118098, 472392, 944784, 3779136, 7558272, 30233088, 120932352, 241864704, 967458816, 1934917632.
The column labeled '1' of least occurrences of a width pattern of length 2k-1 is sequence A038547: least number with exactly k odd divisors.

Cf. A235791, A237048, A237270, A237591, A237593, A249223, A250071, A262045, A318843, A341969, A342592, A342594, A342595, A342596, A367377.

t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]]
(* row n in triangle of A249223 *)
t262045[n_] := Join[t249223[n], Reverse[t249223[n]]] (* row n in triangle of A262045 *)
widthPattern[n_] := Map[First, Split[t262045[n]]]
nOddDivs[n_] := Length[Divisors[NestWhile[#/2&, n, EvenQ[#]&]]]
count[n_, k_] := Length[Union[Map[widthPattern, Select[Range[n], nOddDivs[#]==k&]]]]
(* count of distinct width patterns for numbers with k odd divisors in the range 1 .. n *)

cp's OEIS Frontend

A342594 Earliest occurrence of the next distinct width pattern (as listed in A342592) in the symmetric representation of sigma(n) not yet encountered as n increases.

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A342595 Irregular triangle of A342592 read by rows arranged first by length of the width pattern and then lexicographically within blocks of patterns of equal length.

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A342596 Numbers k of the earliest occurrence of widths patterns in the symmetric representation of sigma listed in the ordering of patterns in A342595.

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A367377 Square array T(n, k), n >= 1, k >= 1, read by antidiagonals, of the least numbers whose symmetric representation of sigma instantiate the unimodal width pattern 1, 2, ..., n, ..., 2, 1 repeated k times separated by instances of width 0.

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A370206 Numbers j whose symmetric representation of sigma(j) consists of two copies of unimodal width pattern 121 separated by 0.

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A370205 Numbers j whose symmetric representation of sigma(j) consists of the single unimodal width pattern 121.

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A370209 a(n) is the smallest number of the form 2^k * p * (2^(k+1) * p + 1) where 2 < p < 2^(k+1) is the n-th prime and 2^(k+1) * p + 1 is prime, or -1 if no such number exists.

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A372180 Square array read by antidiagonals upwards in which T(n,m) is the n-th number whose symmetric representation of sigma consists of m copies of unimodal pattern 121 (separated by 0's if m > 1).

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A367370 a(k) is the number of different widths patterns in the symmetric representation of sigma for numbers having k odd divisors.

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