A241008 -id:A241008 - cp's OEIS frontend

A318843 a(n) is the smallest number k such that the symmetric representation of sigma(k) consists of n parts of width 1.

1, 3, 9, 21, 81, 147, 729, 903, 3025, 6875, 59049, 29095, 531441, 171875, 366025, 643885, 43046721, 3511475

Offset: 1

Hartmut F. W. Hoft, Sep 04 2018

The sequence is infinite since, for example, for any n >= 1 the symmetric representation of sigma(3^n) consists of n + 1 parts of width 1. However, it is not increasing since a(11) = 59049 = 3^10 and a(12) = 29095 = 5 * 11 * 23^2. Also a(13) <= 531441 = 3^12.
This sequence is a subsequence of A174905; its subsequences a(n) for odd/even n are subsequences of A241010/A241008, respectively. Some even-indexed elements of this sequence are members of A239663, e.g., a(2), a(4), a(6), a(8) and a(12), but not a(10) = 6875.
The central pair of parts in the symmetric representation of sigma(a(2)), sigma(a(4)) and sigma(a(8)) meets at the diagonal (see A298856).
From Hartmut F. W. Hoft, Oct 04 2021: (Start)
An upper bound to the sequence is a(n) <= 3^(n-1), n >= 1, (see A348171).
For p = 1,2,3,5,7,11,13,17, a(p) = 3^(p-1) and this equality possibly holds for all a(p) with p a prime.
Also, 75 * 10^6 < a(19) <= 3^18, a(20) = 15391255, a(21) = 44289025 and a(n) > 75 * 10^6 for n > 21.
a(13)-a(18) computations based on A348171 rather than A237270.
The symmetric representation of sigma(3^(p-1)), p prime, consists of p parts and its middle part has area 3^((p-1)/2). (End)
a(n) >= A038547(n) with equality for n=1 and primes n since the distinct prime divisors of n can be replaced by primes 3, 5, 7, 11, ... yielding a smaller number k with the same number of odd divisors. However, some parts in the symmetric representation of sigma(k) have width at least 2. - Hartmut F. W. Hoft, Dec 11 2023

			The smallest number k whose symmetric representation of sigma(k) consists of four parts of width one is a(4) = 21. The parts are 11, 5, 5, 11.
a(4) = 3*7 has width pattern, A341969, 1010101 while A038547(4) = 3*5 has width pattern 1012101. a(6) = 3 * 7^2 = 147 has width pattern 10101010101 while A038547(6) = 3^2 * 5 = 45 has width pattern 10121212101. - _Hartmut F. W. Hoft_, Dec 11 2023

Cf. A174905, A237048, A237270, A237271, A237593, A239663, A239665, A240062, A240542, A241008, A241010, A249351, A298856.
Cf. A249223, A348171.
Cf. A038547, A341969.

(* Function path[] is defined in A237270 *)
segmentsSR[pathN0_, pathN1_] := SplitBy[Map[Min, Drop[Drop[pathN0, 1], -1] - pathN1], #==0&]
regions[pathN0_ ,pathN1_] := Select[Map[Apply[Plus, #]&, segmentsSR[pathN0, pathN1]], #!=0&]
width1Q[pathN0_, pathN1_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[pathN0, 1], -1] - pathN1, 1]]]
(* parameter seq is the list of elements of the sequence in interval 1..m-1 already computed with an entry of 0 representing an element not yet found *)
a318843[m_, n_, seq_] := Module[{list=Join[seq, Table[0, 10]], path1=path[m-1], path0, k, a, r, w}, For[k=m, k<=n, k++, path0=path[k]; a=regions[path0, path1]; r=Length[a]; w=width1Q[path0, path1]; If[w && list[[r]]==0, list[[r]]=k]; path1=path0]; list]
a318843[2,60000,{1}] (* data - actually computed in steps *)

a(13)-a(18) from Hartmut F. W. Hoft, Oct 04 2021

A264102 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.

Original entry on oeis.org

21, 27, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 125, 129, 133, 141, 145, 155, 159, 161, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 230, 235, 237, 249, 250, 253, 259, 265, 267, 287, 290, 291, 295, 301, 303, 305, 309, 310, 319, 321, 327, 329, 335

Offset: 1

Hartmut F. W. Hoft, Nov 03 2015

The areas of the first two regions are (2^(m+1) - 1) * (p * q + 1) / 2 and (2^(m+1) - 1) * (p + q) / 2, respectively. Twice their sum equals sigma(n) = (2^(m+1) - 1) * (p + 1) * (q + 1).

For a proof of the formula for this sequence see the link.

			65 = 5*13 is in the sequence since m = 0 and 2 < 5 < 10 < 13. The first two regions in the symmetric representation of sigma(65) = 84 start with legs 1 and 5 of the Dyck path and have areas 33 and 9, respectively.
406 = 2*7*29 is in the sequence since m=1 and 4 < 7 < 28 < 29. The first two regions in the symmetric representation of sigma(406) = 720 start with legs 1 and 7 and have areas 306 and 54, respectively. Note also that 406 is a triangular number and the middle two regions meet at the center of the Dyck path.
One case in the formula for the sequence is the 3-parameter expression n = 2^m * p * q with p and q distinct primes satisfying the stated conditions. That subsequence can be visualized as a skew tetrahedron since the start of each "line" on an irregular "triangular" side of the "tetrahedron" is determined by a different prime number and each layer is determined by a different power of two. Below are the first three layers with primes p designating columns and primes q rows.
m=0| 3    5    7    11   13
-----------------------------
7  | 21
11 | 33   55
13 | 39   65
17 | 51   85   119
19 | 57   95   133
23 | 69   115  161  253
29 | 87   145  203  319  377
31 | 93   155  217  341  403
37 | 111  185  259  407  481
41 | 123  205  287  451  533
...
89 | 267  445  623  979  1157
...
Column 1 is A001748 except for the first three terms and column 2 is A001750 except for the first four terms in the two resepctive sequences.
m=1| 3    5    7    11   13
-------------------------------
23 |     230
29 |     290  406
31 |     310  434
37 |     370  518
41 |     410  574
43 |     430  602
47 |     470  658  1034
53 |     530  742  1166  1378
...
89 |     890  1246 1958  2314
...
m=2| 3    5    7    11   13
-------------------------------
89 |               3916
97 |               4268
101|               4444
103|               4532
107|               4708  5564
109|               4796  5668
...
The fourth layer for m = 3 starts with number 37672 in column p = 17 and row q = 277.
The subsequence of the 2-parameter case n = 2^m * p^3 with 2^(m+1) < p gives rise to the following irregular triangle:
p\m| 0      1       2       3
----------------------------------
3  | 27
5  | 125    250
7  | 343    686
11 | 1331   2662    5324
13 | 2197   4394    8788
17 | 4913   9826    19652   39304
19 | 6859   13718   27436   54872
23 | 12167  24334   48668   97336
29 | 24389  48778   97556   195112
...
The first column in this triangle is A030078 except for the first term and the second column is A172190 except for the first two terms respectively in the two sequences.

Hartmut F. W. Hoft, Diagram of symmetric representations of sigma
Hartmut F. W. Hoft, Proof of formula for 4 regions of width 1

Cf. A001748, A001750, A030078, A172190.
For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.
Subsequence of A280107.

mpStalk[m_, p_, bound_] := Module[{q=NextPrime[2^(m+1)*p], list={}}, While[2^m*p*q<=bound, AppendTo[list, 2^m*p*q]; q=NextPrime[q]]; If[2^m*p^3<=bound, AppendTo[list, 2^m*p^3]]; list]
mTriangle[m_, bound_] := Module[{p=NextPrime[2^(m+1)], list={}}, While[2^m*p*NextPrime[2^(m+1)*p]<=bound, list=Union[list, mpStalk[m, p, bound]]; p=NextPrime[p]]; list]
(* 2^(4m+3)<=bound is a simpler test, but computes some empty stalks *)
a264102[bound_] := Module[{m=0, list={}}, While[2^m*NextPrime[2^(m+1)]*NextPrime[2^(m+1)*NextPrime[2^(m+1)]]<=bound, list=Union[list, mTriangle[m, bound]]; m++]; list]
a264102[335] (* data *)

n = 2^m * p * q where m >= 0, p > 2 is prime, 2^(m+1) < p < 2^(m+1) * p < q, and either q is prime or q = p^2.

A357581 Square array read by antidiagonals of numbers whose symmetric representation of sigma consists only of parts that have width 1; column k indicates the number of parts and row n indicates the n-th number in increasing order in each of the columns.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 8, 7, 25, 21, 16, 10, 49, 27, 81, 32, 11, 50, 33, 625, 147, 64, 13, 98, 39, 1250, 171, 729, 128, 14, 121, 51, 2401, 207, 15625, 903, 256, 17, 169, 55, 4802, 243, 31250, 987, 3025, 512, 19, 242, 57, 14641, 261, 117649, 1029, 3249, 6875

Offset: 1

Hartmut F. W. Hoft, Oct 04 2022

This sequence is a permutation of A174905. Numbers in the even numbered columns of the table form A241008 and those in the odd numbered columns form A241010. The first row of the table is A318843.

This sequence is a subsequence of A240062 and each column in this sequence is a subsequence in the respective column of A240062.

			The upper left hand 11 X 11 section of the table for a(n) <= 2*10^7:
     1   2    3   4      5    6         7     8      9     10        11 ...
  ----------------------------------------------------------------------
     1   3    9  21     81  147       729   903   3025   6875     59049
     2   5   25  27    625  171     15625   987   3249   7203   9765625
     4   7   49  33   1250  207     31250  1029   4761  13203  19531250
     8  10   50  39   2401  243    117649  1113   6561  13527       ...
    16  11   98  51   4802  261    235298  1239   7569  14013       ...
    32  13  121  55  14641  275   1771561  1265   8649  14499       ...
    64  14  169  57  28561  279   3543122  1281  12321  14661       ...
   128  17  242  65  29282  333   4826809  1375  14161  15471       ...
   256  19  289  69  57122  363   7086244  1407  15129  15633       ...
   512  22  338  85  58564  369   9653618  1491  16641  15957       ...
  1024  23  361  87  83521  387  19307236  1533  17689  16119       ...
  ...
Each column k > 1 contains odd and even numbers since, e.g., 5^(k-1) and 2 * 5^(k-1) belong to it.
Column 1: A000079, subsequence of A174973 = A238443, and of column 1 in A240062.
Column 2: A246955, subsequence of A239929; 78 is the smallest number not in A246955.
Column 3: A247687, subsequence of A279102; 15 is the smallest number not in A247687.
  Odd numbers in column 3: A001248(k), k > 1.
Column 4: A264102, subsequence of A280107; 75 is the smallest number not in A264102.
Column 5: subsequence of A320066; 63 = A320066(1) is not in column 5.
  Numbers in column 5 have the form 2^k * p^4 with p > 2 prime and 0 <= k < floor(log_2(p)).
  Odd numbers in column 5: A030514(k), k > 1.
Column 6: subsequence of A320511; 189 is the smallest number not in column 6.
  Smallest even number in column 6 is 5050.
Column 7: Numbers have the form 2^k * p^6 with p > 2 prime and 0 <= k < floor(log_2(p)).
  Odd numbers in column 7: A030516(k), k > 1.
Numbers in the column numbered with the n-th prime p_n have the form: 2^k * p^(p_n - 1) with p > 2 prime and 0 <= k < floor(log_2(p_n)).

Cf. A000079, A001248, A030514, A030516, A174905, A174973, A237593, A238443, A239929, A241008, A241010, A246955, A247687, A264102, A279102, A280107, A318843, A320066, A320511, A341969, A341970, A341971.

(* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
width1Table[n_, {r_, c_}] := Module[{k, list=Table[{}, c], wL, wLen, pCount, colLen}, For[k=1, k<=n, k++, wL=a341969[k]; wLen=Length[wL]; pCount=(wLen+1)/2; If[pCount<=c&&Length[list[[pCount]]]=1, j--, vec[[PolygonalNumber[i+j-2]+j]]=arr[[i, j]]]]; vec]
a357581T[n_, r_] := TableForm[width1Table[n, {r, r}]]
a357581[120000, 10] (* sequence data - first 10 antidiagonals *)
a357581T[120000, 10] (* upper left hand 10x10 array *)
a357581T[20000000, 11] (* 11x11 array - very long computation time *)

A244250 Triangle read by rows in which row n lists the widths in the first octant of the symmetric representation of sigma(n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 26 2014

For the definition of k-th width of the symmetric representation of sigma(n) see A249351.
Row n list the first n terms of the n-th row of A249351.
It appears that the leading 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).
For more information see A237591, A237593.

			Triangle begins:
1;
1, 1;
1, 1, 0;
1, 1, 1, 1;
1, 1, 1, 0, 0;
1, 1, 1, 1, 1, 2;
1, 1, 1, 1, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 0;
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2;
1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0;
...

Cf. A000203, A067742, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A239660, A239932-A239934, A240542, A241008, A241010, A244360, A244361, A245685, A246955, A246956, A247687, A249223, A249351, A250068, A250070, A250071.

A298855 Squarefree semiprimes pq for which the symmetric representation of sigma(pq) has four parts, in increasing order.

Original entry on oeis.org

21, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 129, 133, 141, 145, 155, 159, 161, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 235, 237, 249, 253, 259, 265, 267, 287, 291, 295, 301, 303, 305, 309, 319, 321, 327, 329, 335, 339, 341, 355, 365, 371, 377, 381, 393, 395

Offset: 1

Hartmut F. W. Hoft, Jan 27 2018

All numbers in this sequence are odd since the symmetric representation of 2*p, p prime > 3, has two parts each of size 3*(p+1)/2, and that for 6 has one part of size 12.
A number in this sequence has the form p*q, p and q prime, 3 <= p and 2*p < q, since in this case 2*p <= floor((sqrt(8*p*q + 1) - 1)/2) < q so that 1's in row p*q of A237048 occur only in positions 1, 2, p and 2*p.
This sequence is a subsequence of A046388, hence of A006881, as well as of A174905, A241008 and A280107.
The two central parts of the symmetric representation of sigma(p*q), each of size (p+q)/2, meet on the diagonal when q = 2*p + 1 since in this case 2*p = floor((sqrt(8*p*q + 1) - 1)/2). These triangular numbers p*(2p+1) form sequence A156592, except for its first element 10, and form a subsequence of the diagonal in the associated irregular triangle of this sequence given in the Example section. They also are a subsequence of A264104. A function to compute the coordinates on the diagonal where the two central parts meet is defined in sequence A240542.
Except for missing 10 the intersection of this sequence and A298856 equals A156592.

			21=3*7 is the smallest number in the sequence since 2*3<7.
1081=23*(2*23+1) is in the sequence; its central parts meet at 751 on the diagonal.
The semiprimes p*q can be arranged as an irregular triangle with rows and columns labeled by the respective odd primes:
  q\p|   3    5    7   11   13   17   19   23
  ---+---------------------------------------
   7 |  21
  11 |  33   55
  13 |  39   65
  17 |  51   85  119
  19 |  57   95  133
  23 |  69  115  161  253
  29 |  87  145  203  319  377
  31 |  93  155  217  341  403
  37 | 111  185  259  407  481  629
  41 | 123  205  287  451  533  697  779
  43 | 129  215  301  473  559  731  817
  47 | 141  235  329  517  611  799  893 1081

Cf. A001358, A005384, A005385, A006881, A046388, A068443, A156592, A174905, A237048, A237270, A237593, A240542, A241008, A264104, A280107, A298856.

(* Function a237270[] is defined in A237270 *)
a006881Q[n_] := Module[{f=FactorInteger[n]}, Length[f]==2 && AllTrue[Last[Transpose[f]], #==1&]]
a298855[m_, n_] := Select[Range[m, n], a006881Q[#] && Length[a237270[#]]==4 &]
a298855[1, 400] (* data *)
(* column for prime p through number n *)
stalk[n_, p_] := Select[a298855[1, n], First[First[FactorInteger[#]]]==p&]

A362866 Numbers k with the property that the parts of the symmetric representation of sigma(k) are two octagons.

Original entry on oeis.org

10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 1

Omar E. Pol, May 06 2023

nonn

Note that odd primes (A065091) are also the numbers j with the property that the parts of the symmetric representation of sigma(j) are two rectangles or more generally two quadrilaterals.
Conjecture 1: The octagons are S-shaped and they have width 1.
Conjecture 2: This sequence is also the primes doubled (or even semiprimes) >= 10 (Cf. A100484). - Omar E. Pol, Aug 15 2023
For the symmetric representation of sigma(n) to consist of 2 octagons the first 3 entries in row n of the triangle of A249223 must be nonzero, hence must be 1's, indicating width 1, with the remaining entries zero. Therefore, row n of A237048 is 100100..., implying n = 2*p with p>3 prime. Both conjectures are true. - Hartmut F. W. Hoft, Aug 22 2023
From Omar E. Pol, Aug 23 2023: (Start)
Also the row numbers of the triangle A364639 where the rows are [1, 0, -1, 1] or where the rows start with [1, 0, -1, 1] and the remaining terms are zeros.
Each supersequence A063221 >= 10 and A091999 >= 10 gives the numbers k with the property that the first part of the symmetric representation of sigma(k) is an  octagon. In that case each supersequence gives the row numbers of the triangle A364639 where the rows start with [1, 0, -1]. (End)

			The symmetric representation of sigma(14) in the first quadrant is as follows:
.   _ _ _ _ _ _ _ _
   |_ _ _ _ _ _ _  |
                 | |
                 | |_
                 |_ _|
                     |_ _
                       | |_ _ _
                       |_ _ _  |
                             | |
                             | |
                             | |
                             | |
                             | |
                             | |
                             |_|
.
The diagram has only two parts (or polygons) and both are octagons so 14 is in the sequence.

Subsequence of A063221, A091999, A100484, A239929 and possibly more.

Cf. A000203, A001747, A065091, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A241008, A245092, A246955, A249223, A249351, A262626, A364414, A364639, A365081.

More terms from Omar E. Pol, Aug 15 2023

A264104 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one and two regions meet at the center of the Dyck path.

Original entry on oeis.org

21, 55, 253, 406, 1081, 1378, 1711, 3403, 3916, 5671, 9316, 11026, 13861, 14878, 15931, 25651, 27028, 34453, 36046, 42778, 50086, 60031, 64261, 73153, 75466, 108811, 114481, 126253, 129286, 154846, 158203, 161596, 171991, 175528, 212878, 258121, 298378, 317206, 326836, 351541, 366796, 371953, 392941

Offset: 1

Hartmut F. W. Hoft, Nov 03 2015

This sequence is a subsequence of A264102 and also of A014105, the second hexagonal numbers. Every number in this sequence is a triangular number.
The sequence A156592 of products of a Sophie Germain prime (A005384) and its associated safe prime (A005385) except for the first pair (2, 5) forms a subsequence of this sequence, the first column in the irregular triangular grid in the example.
The areas of the first two regions are (2^(m+1) - 1) * (2^(m+1) * p^2 * p + 1) / 2 and (2^(m+1) - 1) * (2^(m+1) * p + p + 1) / 2, respectively. Twice their sum equals sigma(n) = (2^(m+1) - 1) * (p + 1) * (2^(m+1) * p + 2).
For a proof of the formula for this sequence see the link.

			406 = 2*7*29 is in the sequence since m = 1 and 4 < 7 < 28 < 29. The first two regions in the symmetric representation of sigma(406) = 720 start with legs 1 and 7 and have areas 306 and 54, respectively. Note also that 406 is a triangular number and the middle two regions meet at the center of the Dyck path.
10 does not belong to this sequence since the symmetric representation of sigma(10) has two regions of width 1 that meet at the diagonal.
There is a natural arrangement of the numbers n = 2^m * p * (2^(m+1) * p + 1) as a sparse irregular triangular (p,m)-grid.
p\m| 0      1       2        3        4        5   ...
-------------------------------------------------------
3  | 21
5  | 55
7  |        406
11 | 253            3916
13 |        1378
17 |                9316
19 |
23 | 1081
29 | 1711           27028
31 |
37 |        11026           175528
41 | 3403
43 |        14878
47 |
53 | 5671                           1439056
59 |                                1783216
61 |                        476776
67 |        36046                            9195616
71 |                161596          2582128
73 |        42778                            10916128
...
The first number in the m = 6 column is 181880128 = 2^6*149*19073 in row p = 149 and the second is 228477376 = 2^6*167*21377 in row p = 167.

Hartmut F. W. Hoft, Diagram of symmetric representations of sigma(n), for n = 21, 55, 253, 406
Hartmut F. W. Hoft, Proof of 4 regions width 1 and 2 meet at center

Cf. A005384, A005385, A014105, A156592, A264102.

For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.

mStalk[m_, bound_] := Module[{p=NextPrime[2^(m+1)], list={}}, While[2^m*p*(2^(m+1)*p+1)<=bound, If[PrimeQ[2^(m+1)*p+1], AppendTo[list, 2^m *p*(2^(m+1)*p+1)]]; p=NextPrime[p]]; list]
a264104[bound_] := Module[{m=0, list={}}, While[2^m*NextPrime[2^(m+1)]*(2^(m+1)*NextPrime[2^(m+1)]+1)<=bound, list=Union[list, mStalk[m, bound]]; m++]; list]
a264104[400000] (* data *)

n = 2^m * p * (2^(m+1) * p + 1) where m >= 0, 2^(m+1) < p and p as well as 2^(m+1) * p + 1 are prime.

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)

A368949 Complement of A368087 in A174905.

Original entry on oeis.org

21, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 129, 133, 141, 145, 147, 155, 159, 161, 171, 177, 183, 185, 201, 203, 205, 207, 213, 215, 217, 219, 230, 235, 237, 249, 253, 259, 261, 265, 267, 275, 279, 287, 290, 291, 295, 301, 303, 305, 309, 310, 319, 321, 327, 329, 333, 335, 339, 341

Offset: 1

Hartmut F. W. Hoft, Jan 10 2024

nonn

A298855 is a subsequence, number 147 = 3 * 7^2 is the first entry in A368949 not in A298855. This sequence is a subsequence of A174905 = A241008 union A241010.

Alternate definition: Every number n in this sequence has at least 2 distinct odd prime factors and its symmetric representation of sigma consists only of parts of width 1.

			21 is the smallest number in this sequence since 3 * 5 is not in A174905.
The smallest number in this sequence with 3 distinct odd prime factors is 903 = 3*7*43 which has width pattern (A341969) 101010101010101 of length 17 since ... < d_i < 2 * d_i < d_(i+1) < 2 * d_(i+1) ... holds for all its divisors.

Cf. A005279, A174905, A235791, A237048, A237270, A237593, A241008, A241010, A249223, A298855, A341969, A368087.

(* a174905[ ] is defined in A174905 and propQ[ ] in A368087 *)
a368949[m_, n_] := Select[Range[m, n], a174905[#]&&!propQ[#]&]
a368949[1, 350]

A368950 Numbers with 11 odd divisors.

Original entry on oeis.org

59049, 118098, 236196, 472392, 944784, 1889568, 3779136, 7558272, 9765625, 15116544, 19531250, 30233088, 39062500, 60466176, 78125000, 120932352, 156250000, 241864704, 282475249, 312500000, 483729408, 564950498, 625000000, 967458816, 1129900996, 1250000000, 1934917632

Offset: 1

Hartmut F. W. Hoft, Jan 10 2024

nonn

Every number in this sequence has the form 2^k * p^10, k >= 0, where p is an odd prime. Exactly 11 different width patterns (A341969) of the symmetric representation of sigma are instantiated by the numbers in this sequence. The width pattern becomes unimodal for k >= floor(log_2(p^10)), see A367370 and A367377.

			a(1) = 59049 = 3^10, a(9) = 5^10 = 9765625 is the smallest number with prime factor 5, a(19) = 282475249 is the smallest number with prime factor 7 and a(27) = 2^floor(log_2(3^10)) * 3^10 = 32768 * 59049 = 1934917632 is the smallest whose width pattern of its symmetric representation of sigma is unimodal.

Cf. A267983 (lists the sequences of numbers with 1 .. 10 odd divisors), A367370, A367377.

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

N:= 10^10: # for terms <= N
R:= NULL: p:= 2:
do
  p:= nextprime(p);
  if p^10 > N then break fi;
  R:= R, seq(2^i*p^10, i = 0 .. floor(log[2](N/p^10)))
od:
sort([R]); # Robert Israel, Jan 16 2024

numL[p_, b_] := Map[2^# p^10&, Range[0, Floor[Log[2, b/p^10]]]]
primeL[b_] := Most[NestWhileList[NextPrime[#]&, 3, #^10<=b&]]
a368950[b_] := Union[Flatten[Map[numL[#, b]&, primeL[b]]]]
a368950[2 10^9]

cp's OEIS Frontend

A318843 a(n) is the smallest number k such that the symmetric representation of sigma(k) consists of n parts of width 1.

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A264102 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one.

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A357581 Square array read by antidiagonals of numbers whose symmetric representation of sigma consists only of parts that have width 1; column k indicates the number of parts and row n indicates the n-th number in increasing order in each of the columns.

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A244250 Triangle read by rows in which row n lists the widths in the first octant of the symmetric representation of sigma(n).

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A298855 Squarefree semiprimes p*q for which the symmetric representation of sigma(p*q) has four parts, in increasing order.

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A362866 Numbers k with the property that the parts of the symmetric representation of sigma(k) are two octagons.

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A264104 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one and two regions meet at the center of the Dyck path.

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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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A368949 Complement of A368087 in A174905.

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A298855 Squarefree semiprimes pq for which the symmetric representation of sigma(pq) has four parts, in increasing order.