A341969 -id:A341969 - cp's OEIS frontend

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

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.

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]

A320521 a(n) is the smallest even number k such that the symmetric representation of sigma(k) has n parts.

Original entry on oeis.org

2, 10, 50, 230, 1150, 5050, 22310, 106030, 510050, 2065450, 10236350

Offset: 1

Omar E. Pol, Oct 14 2018

It appears that a(n) = 2 * q where q is odd and that the symmetric representation of sigma(a(n)/2) has the same number of parts as that for a(n). Number a(12) > 15000000. - Hartmut F. W. Hoft, Sep 22 2021

			a(1) = 2 because the second row of A237593 is [2, 2], and the first row of the same triangle is [1, 1], therefore between both symmetric Dyck paths there is only one part: [3], equaling the sum of the divisors of 2: 1 + 2 = 3. See below:
.
.     _ _ 3
.    |_  |
.      |_|
.
.
a(2) = 10 because the 10th row of A237593 is [6, 2, 1, 1, 1, 1, 2, 6], and the 9th row of the same triangle is [5, 2, 2, 2, 2, 5], therefore between both symmetric Dyck paths there are two parts: [9, 9]. Also there are no even numbers k < 10 whose symmetric representation of sigma(k) has two parts. Note that the sum of these parts is 9 + 9 = 18, equaling the sum of the divisors of 10: 1 + 2 + 5 + 10 = 18. See below:
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.
a(3) = 50 because the 50th row of A237593 is [26, 9, 4, 3, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 3, 4, 9, 26], and the 49th row of the same triangle is [25, 9, 5, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 3, 5, 9, 25], therefore between both symmetric Dyck paths there are three parts: [39, 15, 39]. Also there are no even numbers k < 50 whose symmetric representation of sigma(k) has three parts. Note that the sum of these parts is 39 + 15 + 39 = 93, equaling the sum of the divisors of 50: 1 + 2 + 5 + 10 + 25 + 50 = 93. (The diagram of the symmetric representation of sigma(50) = 93 is too large to include.)

Row 1 of A320537.
Cf. A237270 (the parts), A237271 (number of parts), A174973 (one part), A239929 (two parts), A279102 (three parts), A280107 (four parts), A320066 (five parts), A320511 (six parts).
Cf. A000203, A018262, A005843, A196020, A235791, A236104, A237048, A237591, A237593, A239663, A239665, A240062, A245092, A262626, A296508.
Cf. A250070, A262045, A341969, A341970, A341971, A347979.

(* support functions are defined in A341969, A341970 & A341971 *)
a320521[n_, len_] := Module[{list=Table[0, len], i, v}, For[i=2, i<=n, i+=2, v=Count[a341969[i], 0]+1;If[list[[v]]==0, list[[v]]=i]]; list]
a320521[15000000,11] (* Hartmut F. W. Hoft, Sep 22 2021 *)

a(6)-a(11) from Hartmut F. W. Hoft, Sep 22 2021

A320537 Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.

Original entry on oeis.org

2, 4, 10, 6, 14, 50, 8, 22, 70, 230, 12, 26, 98, 250, 1150, 16, 34, 110, 290, 1250, 5050, 18, 38, 130, 310, 1450, 5150, 22310, 20, 44, 154, 370, 1550, 5290, 23230, 106030, 24, 46, 170, 406, 1850, 5350, 23690, 106490, 510050, 28, 52, 182, 410, 2030, 5450, 24610, 107410, 513130, 2065450

Offset: 1

Omar E. Pol, Oct 15 2018

This is a permutation of the positive even numbers (A299174).
The union of all odd-indexed columns gives A319796, the even numbers in A071562.
The union of all even-indexed columns gives A319802, the even numbers in A071561.

			From _Hartmut F. W. Hoft_, Oct 06 2021: (Start)
The 10x10 section of table T(n,k):
(Table with first 20 terms from _Omar E. Pol_)
------------------------------------------------------------------
n\k | 1   2   3    4    5     6     7      8       9       10  ...
------------------------------------------------------------------
  1 | 2   10  50   230  1150  5050  22310  106030  510050  2065450
  2 | 4   14  70   250  1250  5150  23230  106490  513130  2115950
  3 | 6   22  98   290  1450  5290  23690  107410  520150  2126050
  4 | 8   26  110  310  1550  5350  24610  110170  530150  2157850
  5 | 12  34  130  370  1850  5450  25070  112010  530450  2164070
  6 | 16  38  154  406  2030  5650  25250  112930  532450  2168150
  7 | 18  44  170  410  2050  5750  25750  114770  534290  2176550
  8 | 20  46  182  430  2150  6250  25990  115690  537050  2186650
  9 | 24  52  190  434  2170  6350  26450  116150  540350  2216950
  10| 28  58  238  470  2350  6550  26750  117070  544870  2219650
   ... (End)

Row 1 is A320521.
Column 1 gives A174973 = A238443, without the 1.
Column 2 gives A244894.
Cf. A000203, A071561, A071562, A236104, A237270, A237271, A237593, A238443, A239663, A239665, A239929, A240062, A245092, A262626, A299174, A319796, A319802, A341969, A341970, A341971, A346969, A348171.

(* function a341969 is defined in A341969 *)
sArray[b_, pMax_] := Module[{list=Table[{}, pMax], i, p}, For[i=2, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax&&Length[list[[p]]]Hartmut F. W. Hoft, Oct 06 2021 *)

Terms a(21) and beyond from Hartmut F. W. Hoft, Oct 06 2021

A346969 1 together with the square array T(n,k) read by upward antidiagonals in which T(n, k), n >= 1, is the n-th odd number j >= 3 such that the symmetric representation of sigma of j has k >= 2 parts.

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 21, 11, 25, 27, 63, 13, 35, 33, 81, 147, 17, 45, 39, 99, 171, 357, 19, 49, 51, 117, 189, 399, 903, 23, 77, 55, 153, 207, 441, 987, 2499, 29, 91, 57, 165, 243, 483, 1029, 2709, 6069, 31, 121, 65, 195, 261, 513, 1113, 2793, 6321, 13915, 37, 135, 69, 231, 275, 567, 1197, 2961, 6325, 14847, 29095

Offset: 1

Hartmut F. W. Hoft, Oct 06 2021

This sequence is a permutation of the odd positive integers.

The first row of table T(n,k) preceded by a(1) = 1 is A239663; the first column is the sequence A065091 of odd primes; the second column contains the squares of the odd primes as a subsequence (see also A247687).

			The 10x10 initial submatrix of table T(n,k):
n\k | 2   3    4    5     6     7     8      9      10     11  ...
------------------------------------------------------------------
  1 | 3   9    21   63    147   357   903    2499   6069   13915
  2 | 5   15   27   81    171   399   987    2709   6321   14847
  3 | 7   25   33   99    189   441   1029   2793   6325   15125
  4 | 11  35   39   117   207   483   1113   2961   6783   15141
  5 | 13  45   51   153   243   513   1197   3025   6875   15351
  6 | 17  49   55   165   261   567   1239   3087   6909   15729
  7 | 19  77   57   195   275   609   1265   3249   7011   16023
  8 | 23  91   65   231   279   621   1281   3339   7203   16611
  9 | 29  121  69   255   297   651   1375   3381   7353   16779
  10| 31  135  75   273   333   729   1407   3591   7581   17157
   ...
a(9) = 25 = T(3,3) since only 9 and 15 are smaller odd numbers whose symmetric representation of sigma consists of three parts. All 3 parts of the symmetric representation of sigma for 9 and for 25 have width 1 while the center part for that of 15 has width 2.

Cf. A065091, A237270, A237271, A237593, A239663, A247687, A320537, A341969, A341970, A341971, A348171.

(* function a341969 is defined in A341969 *)
sArray[b_, pMax_] := Module[{list=Table[{}, pMax+1], i, p}, For[i=3, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax+1&&Length[list[[p]]]

A348142 Square array read by upward antidiagonals in which T(w,p) is the smallest number k whose symmetric representation of sigma(k) consists of p parts with maximum width w occurring in everyone of its p parts.

Original entry on oeis.org

1, 6, 3, 60, 78, 9, 120, 7620, 1014, 21, 360, 28920, 967740, 12246, 81, 840, 261720, 6969720, 116136420, 171366, 147, 3360, 1422120

Offset: 1

Hartmut F. W. Hoft, Oct 04 2021

It appears that the first row is A318843 and that the first column is A250070.
Columns 1 and 2 both are identical with those of the table in A348171 and row 1 is identical with that of A348171.
In the remainder of the 7th antidiagonal a(24..26) > 120*10^6, a(27) = 1922622, and a(28) = 903.

			The 10x8 section of the table T(w,p) with dashes indicating values greater than 120*10^6; rows w denote the common maximum width in all parts and columns p the number of parts in the symmetric representation of sigma(T(w,p)).
w\p | 1     2         3        4         5        6        7        8  ...
--------------------------------------------------------------------------
  1 | 1     3         9        21        81       147     729      903
  2 | 6     78        1014     12246     171366   1922622 28960854  -
  3 | 60    7620      967740   116136420   -         -       -
  4 | 120   28920     6969720    -
  5 | 360   261720      -
  6 | 840   1422120     -
  7 | 3360  22622880    -
  8 | 2520  12728520    -
  9 | 5040  50858640    -
  10| 10080    -
   ...
The symmetric representation of sigma for T(2,2) = 78 consists of the two parts (84, 84) of maximum widths (2, 2), and that of T(2,3) = 1014 consists of the three parts (1020, 156, 1020) of maximum widths (2, 2, 2).

Cf. A237048, A237270, A237271, A237591, A237593, A238443, A239663, A249223, A250070, A262045, A318843, A341969, A341970, A341971, A347979, A347980, A348171.

(* function a341969 is defined in A341969 *)
a348142[n_, {w_, p_}] := Module[{list=Table[0, {i, w}, {j, p}], k, s, c, u}, Monitor[For[k=1, k<=n, k++, s=Map[Max, Select[SplitBy[a341969[k], #!=0&], #[[1]]!=0&]]; c=Length[s]; u=Union[s]; If[Length[u]==1&&u[[1]]<=w&&c<=p, If[list[[u[[1]], c]]==0, list[[u[[1]], c]]=k]]], list]; list]
table=a348142[120000000, {10, 10}] (* 10x10 table; very long computation time *)
p[n_] := n-row[n-1](row[n-1]+1)/2
w[n_] := row[n-1]-p[n]+2
Map[table[[w[#], p[#]]]&,  Range[23]] (* sequence data *)

a((w+p-2)(w+p-1)/2 + p) = T(w,p), for all w, p >= 1.

T(w(n),p(n)) = a(n), for all n >= 1, where p(n) = n - r(n-1) * (r(n-1) + 1)/2, w(n) = r(n-1) - p(n) + 2, and r(n) = floor((sqrt(8*n+1) - 1)/2).

A368087 Numbers of the form 2^k * p^s with k>=0, s>=0, p>2 prime and 2^(k+1) < p.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 26, 27, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 49, 50, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 81, 82, 83, 86, 89, 92, 94, 97, 98, 101, 103, 106, 107, 109, 113, 116, 118, 121, 122, 124, 125, 127, 128

Offset: 1

Hartmut F. W. Hoft, Dec 11 2023

nonn

This sequence is a subsequence of A174905 = A241008 union A241010. The symmetric representation of sigma (cf. A237593) for a number m in this sequence consists of s+1 parts, the number of odd divisors of m, each part having width 1.

			14 = 2*7 is a term since 4 < 7.
44 = 4*11 is a term since 8 < 11.

Cf. A005279, A174905, A235791, A237048, A237270, A237593, A241008, A241010, A249223, A341969.

propQ[n_] := Module[{fL=FactorInteger[n]}, Length[fL]==1||(Length[fL]==2&&fL[[1, 1]]==2&&fL[[1, 1]]^(fL[[1, 2]]+1)

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.

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

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

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A320521 a(n) is the smallest even number k such that the symmetric representation of sigma(k) has n parts.

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A320537 Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.

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A346969 1 together with the square array T(n,k) read by upward antidiagonals in which T(n, k), n >= 1, is the n-th odd number j >= 3 such that the symmetric representation of sigma of j has k >= 2 parts.

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A348142 Square array read by upward antidiagonals in which T(w,p) is the smallest number k whose symmetric representation of sigma(k) consists of p parts with maximum width w occurring in everyone of its p parts.

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A368087 Numbers of the form 2^k * p^s with k>=0, s>=0, p>2 prime and 2^(k+1) < p.

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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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