A247687 -id:A247687 - cp's OEIS frontend

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.

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.

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 *)

A263951 Square numbers in A070552.

Original entry on oeis.org

9, 25, 121, 361, 841, 3481, 3721, 5041, 6241, 10201, 17161, 19321, 32761, 39601, 73441, 121801, 143641, 167281, 201601, 212521, 271441, 323761, 326041, 398161, 410881, 436921, 546121, 564001, 674041, 776161, 863041, 982081, 1062961, 1079521, 1104601, 1142761, 1190281, 1274641, 1324801

Offset: 1

Zak Seidov, Oct 30 2015

nonn

All terms are == 1 (mod 8). For n > 2, a(n) == 1 (mod 120).

This sequence is a subsequence of A247687 and it contains 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). The sequence of those primes p is A048161. Cf. A237593. - Hartmut F. W. Hoft, Aug 06 2020

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Subsequence of A070552.

Cf. A048161, A067755, A068485, A237593, A247687, A263990.

a263951[n_] := Select[Map[Prime[#]^2&, Range[n]], PrimeQ[(#+1)/2]&]
a263951[190] (* Hartmut F. W. Hoft, Aug 06 2020 *)

forprime(p=3, 2000, if(isprime((p^2+1)/2), print1(p^2, ", "))) \\ Altug Alkan, Oct 30 2015

a(n) = A048161(n)^2.
From Hartmut F. W. Hoft, Aug 06 2020: (Start)
a(n) = 2 * A067755(n) + 1, n >= 1.
a(n+2) = 120 * A068485(n) + 1, n >= 1.  (End)

A377654 Numbers m^2 for which the center part (containing the diagonal) of its symmetric representation of sigma, SRS(m^2), has width 1 and area m.

Original entry on oeis.org

1, 9, 25, 49, 81, 121, 169, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1369, 1521, 1681, 1849, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 4225, 4489, 4761, 5041, 5329, 6241, 6561, 6889, 7225, 7569, 7921, 8649, 9025, 9409, 10201, 10609, 11449, 11881, 12321, 12769, 13225, 14161, 14641, 15129, 15625

Offset: 1

Hartmut F. W. Hoft, Nov 03 2024

nonn

Since for numbers m^2 in the sequence the width at the diagonal of SRS(m^2) is 1, the area m of its center part is odd so that this sequence is a proper subsequence of A016754 and since SRS(m^2) has an odd number of parts it is a proper subsequence of A319529. The smallest odd square not in this sequence is 225 = 15^2.  SRS(225) is {113, 177, 113}, its center part has maximum width 2, its width at the diagonal is 1.
The k+1 parts of SRS(p^(2k)), p an odd prime and k >= 0, through the diagonal including the center part have areas (p^(2k-i) + p^i)/2 for 0 <= i <= k. They form a strictly decreasing sequence. Since p^(2k) has 2k+1 divisors and SRS(p^(2k)) has 2k+1 parts, all of width 1 (A357581), the even powers of odd primes form a proper subsequence of A244579. For the subsequence of squares of odd primes p, SRS(p^2) consists of the 3 parts { (p^2 + 1)/2, p, (p^2 + 1)/2 } see A001248, A247687 and A357581.
The areas of the parts of SRS(m^2) need not be in descending order through the diagonal as a(112) = 275^2 = 75625 with SRS(75625) = (37813, 7565, 3443, 1525, 715, 738, 275, 738, 715, 1525, 3443, 7565, 37813) demonstrates.
An equivalent description of the sequence is: The center part of SRS(m^2) has width 1, m is odd, and A249223(m^2, m-1) = 0.
Conjectures (true for all a(n) <= 10^8):
(1) The central part of SRS(a(n)) is the minimum of all parts of SRS(a(n)), 1 <= n.
(2) The terms in this sequence are the squares of the terms in A244579.

			The center part of SRS(a(3)) = SRS(25) has area 5, all 3 parts have width 1, and 25 with 3 divisors also belongs to A244579.
The center part of SRS(a(7)) = SRS(169) has area 13, all 3 parts have width 1, and 169 with 3 divisors also belongs to A244579.
The center part of SRS(a(10)) = SRS(441) has area 21 and width 1, but the maximum width of SRS(441) is 2. Number 441 has 9 divisors and SRS(441) has 7 parts while 21 has 4 divisors and SRS(21) has 4 parts so that 21 is in A244579 while 441 is not.

Cf. A001248, A030514, A030516, A030629, A030631, A179645 (even prime power sequences).

Cf. A003056, A016754, A237591, A237593, A244579, A247687, A249223, A319529, A341969, A357581.

(* t237591 and partsSRS compute rows in A237270 and A237591, respectively *)
(* t249223 and widthPattern are also defined in A376829 *)
row[n_] := Floor[(Sqrt[8 n+1]-1)/2]
t237591[n_] := Map[Ceiling[(n+1)/#-(#+1)/2]-Ceiling[(n+1)/(#+1)-(#+2)/2]&, Range[row[n]]]
partsSRS[n_] := Module[{widths=t249223[n], legs=t237591[n], parts, srs}, parts=widths legs; srs=Map[Apply[Plus, #]&, Select[SplitBy[Join[parts, Reverse[parts]], #!=0&], First[#]!=0&]]; srs[[Ceiling[Length[srs]/2]]]-=Last[widths]; srs]
t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, row[n]]]
widthPattern[n_] := Map[First, Split[Join[t249223[n], Reverse[t249223[n]]]]]
centerQ[n_] := Module[{pS=partsSRS[n]}, Sqrt[n]==pS[[(Length[pS]+1)/2]]]/;OddQ[n]
widthQ[n_] := Module[{wP=SplitBy[widthPattern[n], #!=0&]}, wP[[(Length[wP]+1)/2]]]=={1}/;OddQ[n]
a377654[m_, n_] := Select[Map[#^2&, Range[m, n, 2]], centerQ[#]&&widthQ[#]&]/;OddQ[m]
a377654[1, 125]

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.

A338486 Numbers n whose symmetric representation of sigma(n) consists of 3 regions with maximum width 2.

Original entry on oeis.org

15, 35, 45, 70, 77, 91, 110, 130, 135, 143, 154, 170, 182, 187, 190, 209, 221, 225, 238, 247, 266, 286, 299, 322, 323, 350, 374, 391, 405, 418, 437, 442, 493, 494, 506, 527, 550, 551, 572, 589, 598, 638, 646, 650, 667, 682, 703, 713, 748, 754, 782, 806, 814, 836, 850

Offset: 1

Hartmut F. W. Hoft, Oct 30 2020

nonn

This sequence is a subsequence of A279102. The definition of the sequence excludes squares of primes, A001248, since the 3 regions of their symmetric representation of sigma have width 1 (first column in the irregular triangle of A247687).
Table of numbers in this sequence arranged by the number of prime factors, counting multiplicities:
     2     3     4      5      6      7  ...
  ------------------------------------------
    15    45   135    405   1215   3645
    35    70   225   1125   5625    ...
    77   110   350   1750   8750 744795
    91   130   550   2584    ...    ...
   143   154   572   2750  85455
   187   170   650   3128    ...
   209   182   748   3250
   221   190   836   3496
   247   238   850   3944
   299   266   884   4216
   ...   ...   ...    ...
              1035   9585
               ...    ...
The numbers in the first row of the table above are b(k) = 5*3^k, k>=1, (see A005030) so that infinitely many odd numbers occur outside of the first column. The central region of the symmetric representation of sigma(b(k)) contains 2*k-1 separate contiguous sections consisting of sequences of entire legs of width 2, k>=1 (see Lemma 2 in the link).
Conjecture: The combined extent of these sections in sigma(b(k)) is 2*3^(k-1) - 1 = A048473(k-1), k>=1.
Since each number n in the first column and first row has a prime factor of odd exponent a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 2. For odd numbers n not in the first row or column in which all prime factors have even powers, such as 225 and 5625 in the second row, a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 1 (see Lemma 1 in the link).
For each k>=3 and every prime p such that b(k-1) < 2*p < 4*b(k-2), the odd number p*b(k-1) is in the column of b(k). The two inequalities are equivalent to b(k-1) <= row(p*b(k-1)) < 2*b(k-1) ensuring that the symmetric representation of sigma(p*b(k-1)) consists of 3 regions.
45 is the only odd number in its column (see Lemma 3 in the link).
Since the factors of n = p*q satisfy 2 < p < q < 2*p the first column in the table above is a subsequence of A082663 and of A087718 (see Lemma 4 in the link). Each of the two outer regions consists of a single leg of width 1 and length (1 + p*q)/2. The center region of size p+q consists of two subparts (see A196020 & A280851) of width 1 of sizes 2*p-q and 2*q-p, respectively (see Lemma 5 in the link). The table below arranges the first column in the table above according to the length 2*p-q of their single contiguous extent of width 2 in the center region:
     1     3      5      7      9     11     13     15  ...
------------------------------------------------------
    15    35    187    247    143    391   2257    323
    91    77    493    589    221   1363   3139    437
   703   209    943   2479    551   2911   6649    713
  1891   299   1537   3397    851   3901    ...   1247
  2701   527   4183   8509   1643   6313          1457
  ...    ...    ...    ...    ...    ...          ....
A129521: p*q satisfies 2*p - q = 1 (excluding A129521(1)=6)
A226755: p*q satisfies 2*p - q = 3 (excluding A226755(1)=9)
Sequences with larger differences 2*p - q are not in OEIS.

			a(6) = 91 = 7*13 is in the sequence and in the 2-column of the first table since 1 < 2 < 7 < 13 = row(91) representing the 4 odd divisors 1 - 91 - 7 - 13 (see A237048) results in the following pattern for the widths of the legs (see A249223): 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2. It also is in the 1-column of the second table since it has a single area of width 2 which is 1 unit long.
a(29) = 405 = 5*3^4 is in the sequence and in the 5-column of the first table since 1 < 2 < 3 < 5 < 6 < 9 < 10 < 15 < 18 < 27 = row(405) representing the 10 odd divisors 1 - 405 - 3 - 5 - 135 - 9 - 81 - 15 - 45 - 27 results in the following pattern for the widths of the legs: 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2, and 7 = 2*4 - 1 sections of width 2 in the central region.
a(35) = 506 = 2*11*23 is in the sequence since positions 1 < 4 < 11 < 23 < row(506) = 31 representing the 4 odd divisors 1 - 253 - 11 - 23 results in the following pattern for the widths of the legs: 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2 for 3 regions with width not exceeding 2, with the two outer regions consisting of 3 legs of width 1, and a single area of width 2 in the central region.

Hartmut F. W. Hoft, proofs for quoted lemmas

Cf. A001248, A005030, A048473, A082663, A087718, A129521, A196020, A226755, A235791, A237048, A237270, A237271, A237591, A237593, A247687, A249223, A279102, A280107, A280851.

(* Functions path and a237270 are defined in A237270 *)
maxDiagonalLength[n_] := Max[Map[#[[1]]-#[[2]]&, Transpose[{Drop[Drop[path[n], 1], -1], path[n-1]}]]]
a338486[m_, n_] := Module[{r, list={}, k}, For[k=m, k<=n, k++, r=a237270[k]; If[Length[r]== 3 && maxDiagonalLength[k]==2,AppendTo[list, k]]]; list]
a338486[1, 850]

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]]]

A352696 a(n) = k if the binary representation of k has a 1 (0) exactly where a 1 in the n-th row of A237048 occurs at an odd (even) position, reading from left to right.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 5, 2, 2, 3, 2, 2, 11, 1, 2, 6, 2, 3, 10, 2, 2, 3, 5, 2, 10, 3, 2, 13, 2, 1, 10, 2, 11, 6, 2, 2, 10, 3, 2, 13, 2, 2, 45, 2, 2, 3, 5, 5, 10, 2, 2, 13, 10, 3, 10, 2, 2, 14, 2, 2, 43, 1, 10, 13, 2, 2, 10, 11, 2, 7, 2, 2, 44, 2, 11, 12, 2, 3, 21, 2, 2, 14, 10

Offset: 1

Hartmut F. W. Hoft, Mar 29 2022

nonn

The phrase "symmetric representation of sigma(n)" is abbreviated below as SRS(n).
Every number in this sequence is a nondiving number and therefore in A061854. Number 22 with binary pattern 10110 is the smallest nondiving number in A061854, but not in this sequence since a number n with 5 odd divisors must have the form n = 2^m * p^4 for some prime p and some m>=0, and the pattern 10110 of odd/even positions of 1's in a row of A237048 requires 1's at positions 1 < 2^(m+1) < p < p^2 < 2^(m+1) * p <= row(n), a contradiction.
a(2^n) = 1 for all n>=0. The single part of SRS(2^n) has width 1, see A238443.
a(2^m * p) = 3 for odd primes p < 2^(m+1) with m >= 1. SRS(2^m * p) consists of a single part whose 2 subparts have sizes 2*T(n, 1) - 1 = 2^m * p - 1 and  2*T(n, p) - 1 = 2^m - p where T(n, k) = ceiling((n+1)/k -(k+1)/2), see A235791. The numbers 2^m * p are a subsequence of A174973 = A238443.
a(p^k) = A000975(k+1) for all odd primes p and k >= 0. Number a(p^k) in binary has k+1 digits with 1's and 0's alternating. SRS(p^k) has k+1 parts all of width 1 and of the symmetric sizes T(p^k, p^i) - T(p^k, 2*p^i) = (p^(k-i) + p^i)/2, for 0 <= i <= k. The numbers p^k are a subsequence of A174905, the odd primes p form the 1st column in the irregular triangle of A239929 and the numbers p^2 form the 1st column in the irregular triangle of A247687.

			Sequence values for the first 4 powers of 3: {a(1), a(3), a(9), a(27)} = {1, 2, 5, 10} = {1, 10, 101, 1010}.
Table for a(1..16), a(27) and a(28) together with their lists of the base-2 representation, of the odd/even positions of 1's in the n-th row of A237048, and of the sizes of the parts in SRS(n):
n  a(n) odd/even   A237048         A237270
1   1   {1}        {1}             {1}
2   1   {1}        {1}             {3}
3   2   {1,0}      {1,1}           {2,2}
4   1   {1}        {1,0}           {7}
5   2   {1,0}      {1,1}           {3,3}
6   3   {1,1}      {1,0,1}         {12}
7   2   {1,0}      {1,1,0}         {4,4}
8   1   {1}        {1,0,0}         {15}
9   5   {1,0,1}    {1,1,1}         {5,3,5}
10  2   {1,0}      {1,0,0,1}       {9,9}
11  2   {1,0}      {1,1,0,0}       {6,6}
12  3   {1,1}      {1,0,1,0}       {28}
13  2   {1,0}      {1,1,0,0}       {7,7}
14  2   {1,0}      {1,0,0,1}       {12,12}
15 11   {1,0,1,1}  {1,1,1,0,1}     {8,8,8}
16  1   {1}        {1,0,0,0,0}     {31}
...
27 10   {1,0,1,0}  {1,1,1,0,0,1}   {14,6,6,14}
28  3   {1,1}      {1,0,0,0,0,0,1} {56}
...

Cf. A000975, A061854, A174905, A174973, A235791, A237048, A237270, A237591, A237593, A238443, A239929, A247687, A352030.

(* function a237048[ ] is defined in A237048 *)
b237048[n_] := Fold[2#1+Mod[#2, 2]&, 0, Flatten[Position[a237048[n], 1]]]
a352696[n_] := Map[b237048, Range[n]]
a352696[85]

cp's OEIS Frontend

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

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A263951 Square numbers in A070552.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A377654 Numbers m^2 for which the center part (containing the diagonal) of its symmetric representation of sigma, SRS(m^2), has width 1 and area m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

A338486 Numbers n whose symmetric representation of sigma(n) consists of 3 regions with maximum width 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A352696 a(n) = k if the binary representation of k has a 1 (0) exactly where a 1 in the n-th row of A237048 occurs at an odd (even) position, reading from left to right.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica