A238524 -id:A238524 - cp's OEIS frontend

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Also the rows of both triangles A237270 and A237593 interleaved.
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).
The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.
The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.
Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.
Odd-indexed rows are also the rows of the irregular triangle A237270.
Even-indexed rows are also the rows of the triangle A237593.
Lengths of the odd-indexed rows are in A237271.
Lengths of the even-indexed rows give 2*A003056.
Row sums of the odd-indexed rows gives A000203, the sum of divisors function.
Row sums of the even-indexed rows give the positive even numbers (see A005843).
Row sums give A245092.
From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

			Irregular triangle begins:
  1;
  1, 1;
  3;
  2, 2;
  2, 2;
  2, 1, 1, 2;
  7;
  3, 1, 1, 3;
  3, 3;
  3, 2, 2, 3;
  12;
  4, 1, 1, 1, 1, 4;
  4, 4;
  4, 2, 1, 1, 2, 4;
  15;
  5, 2, 1, 1, 2, 5;
  5, 3, 5;
  5, 2, 2, 2, 2, 5;
  9, 9;
  6, 2, 1, 1, 1, 1, 2, 6;
  6, 6;
  6, 3, 1, 1, 1, 1, 3, 6;
  28;
  7, 2, 2, 1, 1, 2, 2, 7;
  7, 7;
  7, 3, 2, 1, 1, 2, 3, 7;
  12, 12;
  8, 3, 1, 2, 2, 1, 3, 8;
  8, 8, 8;
  8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
  31;
  9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
  ...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
   n  A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   1     1   =      1      |_| | | | | | | | | | | | | | | |
   2     3   =      3      |_ _|_| | | | | | | | | | | | | |
   3     4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
   4     7   =      7      |_ _ _|    _|_| | | | | | | | | |
   5     6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
   6    12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
   7     8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
   8    15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
   9    13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
  10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
  11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
  12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
  13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
  14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
  15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
  16    31   =     31      |_ _ _ _ _ _ _ _ _|
  ...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
.                                 A237593
Level                               _ _
1                                 _|1|1|_
2                               _|2 _|_ 2|_
3                             _|2  |1|1|  2|_
4                           _|3   _|1|1|_   3|_
5                         _|3    |2 _|_ 2|    3|_
6                       _|4     _|1|1|1|1|_     4|_
7                     _|4      |2  |1|1|  2|      4|_
8                   _|5       _|2 _|1|1|_ 2|_       5|_
9                 _|5        |2  |2 _|_ 2|  2|        5|_
10              _|6         _|2  |1|1|1|1|  2|_         6|_
11            _|6          |3   _|1|1|1|1|_   3|          6|_
12          _|7           _|2  |2  |1|1|  2|  2|_           7|_
13        _|7            |3    |2 _|1|1|_ 2|    3|            7|_
14      _|8             _|3   _|1|2 _|_ 2|1|_   3|_             8|_
15    _|8              |3    |2  |1|1|1|1|  2|    3|              8|_
16   |9                |3    |2  |1|1|1|1|  2|    3|                9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n.
The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n).
The total number of vertical line segments in the n-th level of the diagram equals A131507(n).
The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018
The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022

Robert Price, Table of n, a(n) for n = 1..16048 (n=1..412 rows)
Omar E. Pol, An infinite stepped pyramid (A237593, A237270, A262626)
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16)
Index entries for sequences related to sigma(n)

Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871.
Cf. A000079 (powers of 2)........., for the characteristic shape see A346872.
Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level.
Cf. A000217 (triangular numbers).., for the characteristic shape see A346873.
Cf. A000225 (Mersenne numbers)...., for a visualization see A346874.
Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875.
Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876.
Cf. A000668 (Mersenne primes)....., for a visualization see A346876.
Cf. A001097 (twin primes)........., for a visualization see A346871.
Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level.
Cf. A002378 (oblong numbers)......, for a visualization see A346873.
Cf. A008586 (multiples of 4)......, perimeters of the successive levels.
Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613.
Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo.
Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864.
Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level.
Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others).
Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238.
Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.

A237287 Numbers that are not practical: positive integers n such that there exists at least one number k <= sigma(n) that is not a sum of distinct divisors of n.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 102, 103, 105

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

Complement of A005153 (practical numbers).
Numbers n such that A030057(n) < n.
First differs from A237046 at a(48).
First differs from A238524 at a(55). - Omar E. Pol, Mar 09 2014
First differs from A378471 at a(72). - Hartmut F. W. Hoft, Nov 27 2024

			5 is in the sequence because there are 3 numbers <= sigma(5) = 6 that are not a sum of any subset of distinct divisors of 5: 2, 3 and 4.

Jaroslav Krizek, Table of n, a(n) for n = 1..5000

Cf. A000203, A005153, A237046, A237289, A237290, A378471.

from itertools import count, islice
from sympy import factorint
def A237287_gen(startvalue=1): # generator of terms
    for m in count(max(startvalue,1)):
        if m > 1:
            l = (~m & m-1).bit_length()
            if l>0:
                P = (1<>l).items():
                    if p > 1+P:
                        yield m
                        break
                    P *= (p**(e+1)-1)//(p-1)
            else:
                yield m
A237387_list = list(islice(A237287_gen(),30)) # Chai Wah Wu, Jul 05 2023

More terms added by Hartmut F. W. Hoft, Nov 27 2024, in order to show the difference from A378471.

A244579 Numbers k with the property that the number of parts in the symmetric representation of sigma(k) equals the number of divisors of k.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 79, 81, 83, 85, 87, 89, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 145

Offset: 1

Omar E. Pol, Jul 02 2014

nonn

Numbers n such that A243982(n) = 0.
First differs from A151991 at a(25).
Let n = 2^m * q with m >= 0 and q odd. Let c_n denote the count of regions in the symmetric representation of sigma(n), which is determined by the positions of 1's in the n-th row of A237048. The maximum of c_n occurs when odd and even positions of 1's alternate implying that all regions have width 1, denoted by w_n = 1. When m > 0 then sigma_0(n) > sigma_0(q) and c_n = sigma_0(n) is impossible. Therefore, exactly those odd n with w_n = 1 are in this sequence. Furthermore, since the 1's in A237048 represent the odd divisors of n, their odd-even alternation expresses the property 2*f < g for any two adjacent divisors f < g of odd number n; in other words, this sequence is also the complement of A090196 relative to the odd numbers. This last property permits computations of elements in this sequence faster than with function a244579, which is based on Dyck paths. - Hartmut F. W. Hoft, Oct 11 2015
From Hartmut F. W. Hoft, Dec 06 2016: (Start)
Also, integers n such that for any pair a < b of divisors of n the inequality 2*a < b holds (hence n is odd).
Let 1 = d_1 < ... < d_k = n be all (odd) divisors of n. The property 2*d_i < d_(i+1), for 1 <= i < k, is equivalent for the 1's in the n-th row of A249223 to be in positions 1 = d_1 < 2 < d_2 < 2*d_2 < ... < d_i <2*d_i < d_(i+1) < ... where 2*d_i represents the odd divisor e_i with d_i * e_i = n. In other words, the odd divisors are the number of parts in the symmetric representation of sigma(n). The rightmost 1 in the n-th row occurs in an odd (even) position when k is odd (even).
As a consequence this sequence is also the complement of A090196 in the set of odd numbers. (End)

			9 is in the sequence because the parts of the symmetric representation of sigma(9) are [5, 3, 5] and the divisors of 9 are [1, 3, 9] and in both cases there is the same number of elements: A237271(9) = A000005(9) = 3.
See the link for a diagram of the symmetric representations of sigma for sequence data listed above. The symmetric representations of sigma(a(35)) = sigma(81) = sigma(3^4) consists of 5 regions whose areas are [41, 15, 9, 15, 41] and computed as 41 = (3^4+3^0)/2, 15 = (3^3+3^1)/2, and 9 = 3^2 for the central area. Observe also that the 81st row in triangle A237048 is [ 1 1 1 0 0 1 0 0 1 0 0 0 ] with the 1's in positions 1, 2, 3, 6, and 9. This is the largest count for the symmetric regions of sigma shown in the diagram. - _Hartmut F. W. Hoft_, Oct 11 2015

Hartmut F. W. Hoft, Illustration of the symmetric representations of sigma for sequence data

Cf. A000005, A001227, A005279, A071561, A071562, A090196, A000203, A196020, A236104, A237048, A237270, A237271, A237593, A238443, A238524, A239657, A239929, A239663, A240062, A241558, A241559, A243982, A245092.

(* Function a237270[] is defined in A237270 *)
a244579[m_, n_] := Select[Range[m,n], Length[a237270[#]] == Length[Divisors[#]]&]
a244579[1, 150] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)
(* alternative function using the divisor property *)
divisorPairsQ[n_] := Module[{d=Divisors[n]}, Select[2*Most[d] - Rest[d], # >= 0&] == {}]
a244579Alt[m_?OddQ, n_] := Select[Range[m, n, 2], divisorPairsQ]
a244579Alt[1, 145] (* data *)
(* Hartmut F. W. Hoft, Oct 11 2015 *)

A237271(a(k)) = A000005(a(k)).

A244894 Composite numbers n with the property that the symmetric representation of sigma(n) has two parts.

Original entry on oeis.org

10, 14, 22, 26, 34, 38, 44, 46, 52, 58, 62, 68, 74, 76, 78, 82, 86, 92, 94, 102, 106, 114, 116, 118, 122, 124, 134, 136, 138, 142, 146, 148, 152, 158, 164, 166, 172, 174, 178, 184, 186, 188, 194, 202, 206, 212, 214, 218, 222, 226, 232, 236, 244, 246, 248, 254, 258, 262, 268, 274, 278, 282, 284, 292, 296, 298, 302, 314, 316, 318, 326, 328, 332, 334, 344, 346, 348, 354, 356, 358

Offset: 1

Omar E. Pol, Jul 07 2014

nonn

Even numbers in A239929.

By definition the two parts of the symmetric representation of sigma(n) are sigma(n)/2 and sigma(n)/2.

			Illustration of the symmetric representation of sigma(n) in the second quadrant for the first four elements of this sequence: [10, 14, 22, 26].
.
.                             _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                            |  _ _ _ _ _ _ _ _ _ _ _ _ _|
.                            | |
.                            | |
.                            | |  _ _ _ _ _ _ _ _ _ _ _ _
.                      21 _ _| | |  _ _ _ _ _ _ _ _ _ _ _|
.                        |_ _ _| | |
.                     _ _|       | |
.                   _|     18 _ _| |
.                  |         |_ _ _|
.            21 _ _|        _|
.              | |        _|
.     _ _ _ _ _| | 18 _ _|                _ _ _ _ _ _ _ _
.    |  _ _ _ _ _|   | |                 |  _ _ _ _ _ _ _|
.    | |      _ _ _ _| |                 | |
.    | |     |  _ _ _ _|             12 _| |
.    | |     | |                       |_ _|  _ _ _ _ _ _
.    | |     | |                 12 _ _|     |  _ _ _ _ _|
.    | |     | |              _ _ _| |    9 _| |
.    | |     | |             |  _ _ _|  9 _|_ _|
.    | |     | |             | |      _ _| |
.    | |     | |             | |     |  _ _|
.    | |     | |             | |     | |
.    | |     | |             | |     | |
.    | |     | |             | |     | |
.    | |     | |             | |     | |
.    |_|     |_|             |_|     |_|
.
n:    26      22              14      10
.
Sigma(10) =  9 +  9 = 18.
Sigma(14) = 12 + 12 = 24.
Sigma(22) = 18 + 18 = 36.
Sigma(26) = 21 + 21 = 42.
.

R. J. Mathar, Table of n, a(n) for n = 1..999

Cf. A237271 (number of parts), A237270, A237593, A238443, A238524, A239660, A239929, A239932, A239934, A245092, A262626, A280107 (4 parts).

Extended by R. J. Mathar, Oct 04 2018

A279244 Numbers k with the property that both the smallest and the largest Dyck path of the symmetric representation of sigma(k) share some line segments.

Original entry on oeis.org

5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92

Offset: 1

Omar E. Pol, Dec 08 2016

nonn

Numbers k such that the symmetric representation of sigma(k) is formed by more than two parts, or that it is formed by only two parts and they do not meet at the center.
Numbers k whose total length of all line segments of the symmetric representation of sigma(k) is < 4*k (cf. A348705). - Omar E. Pol, Nov 02 2021
a(n) is also 1 plus the n-th term of the complement of A323648. - Hartmut F. W. Hoft, Feb 22 2025

			5, 7, 9, 11, 13, 14, and 15 are in the sequence because the smallest and the largest Dyck path of their symmetric representation of sigma share some line segments, as shown below.
Illustration of initial terms:
n
.              _   _   _   _   _ _ _
.             | | | | | | | | | | | |
.             | | | | | | | | | | | |
.            _|_| | | | | | | | | | |
.      _ _ _|    _|_| | | | | | | | |
5     |_ _ _|  _|  _ _|_| | | | | | |
.      _ _ _ _|  _| |  _ _|_| | | | |
7     |_ _ _ _| |_ _|_|    _ _|_| | |
.      _ _ _ _ _|  _|     |  _ _ _|_|
9     |_ _ _ _ _| |      _|_| |
.      _ _ _ _ _ _|  _ _|    _|
11    |_ _ _ _ _ _| |  _|  _|
.      _ _ _ _ _ _ _| |_ _|
13    |_ _ _ _ _ _ _| |
14    |_ _ _ _ _ _ _ _|
15    |_ _ _ _ _ _ _ _|
...

Complement of A279029.
Indices of positive terms in A279228.
Subsequence of A238524.
Cf. A000203, A008586, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A279029, A279244, A323648, A348705.

(* Function a279029Q is defined in A279029 *)
a279244[n_] := Select[Range[n], !a279029Q[#]&]
a279244[92] (* Hartmut F. W. Hoft, Feb 20 2025 *)

A365434 Irregular triangle read by rows: T(n,k) is the number of cubes in the k-th level of the ziggurat of order n described in A347186, n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 1, 4, 2, 7, 5, 3, 1, 6, 4, 2, 12, 9, 7, 5, 3, 1, 8, 6, 4, 2, 15, 13, 11, 9, 7, 5, 3, 1, 13, 9, 7, 5, 2, 18, 16, 14, 12, 10, 8, 6, 4, 2, 12, 10, 8, 6, 4, 2, 28, 24, 22, 20, 16, 14, 12, 9, 7, 5, 3, 1, 14, 12, 10, 8, 6, 4, 2, 24, 22, 20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 24, 19, 17, 15, 11, 9, 7, 3, 1, 1

Offset: 1

Omar E. Pol, Oct 19 2023

The values of n when the number of terms in row n is equal to n give A174973.
The values of n when the number of terms in row n is not equal to n give A238524.
If and only if n is a power of 2 then row n lists the first n odd numbers in decreasing order.
If and only if n is an odd prime then row n lists the first (n + 1)/2 positive even numbers in decreasing order.
If and only if n is an even perfect number then row n lists 2*n together with the first n - 1 odd numbers in decreasing order.

			Triangle begins:
   1;
   3,  1;
   4,  2;
   7,  5,  3,  1;
   6,  4,  2;
  12,  9,  7,  5,  3,  1;
   8,  6,  4,  2;
  15, 13, 11,  9,  7,  5,  3,  1;
  13,  9,  7,  5,  2;
  18, 16, 14, 12, 10,  8,  6,  4,  2;
  12, 10,  8,  6,  4,  2;
  28, 24, 22, 20, 16, 14, 12,  9,  7,  5,  3, 1;
  14, 12, 10,  8,  6,  4,  2;
  24, 22, 20, 18, 16, 14, 12, 10,  8,  6,  4, 2;
  24, 19, 17, 15, 11,  9,  7,  3,  1,  1;
  31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1;
  ...

Column 1 gives A000203.
Row sums give A347186.
Row lengths give A365433.
Cf. A000079, A000396, A065091, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238524, A245092, A262626, A347263, A347367, A365195.

A320048 One half of composite numbers k with the property that the symmetric representation of sigma(k) has two parts.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 86, 87, 89, 92, 93, 94, 97, 101, 103, 106, 107, 109, 111, 113, 116, 118, 122, 123, 124, 127, 129, 131, 134, 137, 139, 141, 142, 146, 148, 149, 151, 157, 158, 159, 163, 164

Offset: 1

Omar E. Pol, Oct 04 2018

nonn

Also, even numbers of A239929 divided by two.

First differs from A101550 at a(51). - R. J. Mathar, Oct 04 2018

			5 is in the sequence because 10 is a composite number, and the symmetric representation of sigma(10) = 18 has two parts (as shown below), and 10/2 = 5.
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.

Cf. A101550, A237271 (number of parts), A237270, A237593, A238443, A238524, A239929 (two parts), A239660, A239929, A239932, A239934, A240062 (k parts), A244894, A245092, A262626, A280107 (four parts).

a(n) = A244894(n)/2.

A365195 Height of the first staircase of the ziggurat of order n described in A347186.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 8, 5, 9, 6, 12, 7, 12, 8, 16, 9, 18, 10, 20, 11, 18, 12, 24, 13, 21, 14, 28, 15, 30, 16, 32, 17, 27, 18, 36, 19, 30, 20, 40, 21, 42, 22, 42, 23, 36, 24, 48, 25, 39, 26, 49, 27, 54, 28, 56, 29, 45, 30, 60, 31, 48, 32, 64, 33, 66, 34, 63, 35, 54, 36, 72, 37, 57, 38, 70, 39, 77, 40, 80

Offset: 1

Omar E. Pol, Aug 25 2023

nonn

a(n) is also the number of square cells in the first layer of the symmetric representation of sigma(n) from the border to, at most, the axis of symmetry of the diagram.
The values of a(n) where a(n) = n give A174973.
Since this is a supersequence of A174973 so all powers of 2 and all even perfect numbers are in the sequence.
From Omar E. Pol, Oct 22 2023: (Start)
The values of a(n) where a(n) is not equal to n give A238524.
If n is an odd prime then a(n) = (n + 1)/2.
Shares infinitely many terms with A365433 from which first differs at a(15). (End)

Cf. A000079, A000396, A065091, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238524, A262626, A279387, A347186, A347367, A365433, A365434.

a(n) <= n.

a(2*n-1) = n.

More terms from Omar E. Pol, Oct 22 2023

A365433 Height of the ziggurat of order n described in A347186.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 8, 5, 9, 6, 12, 7, 12, 10, 16, 9, 18, 10, 20, 13, 18, 12, 24, 13, 21, 16, 28, 15, 30, 16, 32, 19, 27, 21, 36, 19, 30, 22, 40, 21, 42, 22, 42, 40, 36, 24, 48, 25, 39, 28, 49, 27, 54, 36, 56, 31, 45, 30, 60, 31, 48, 36, 64, 41, 66, 34, 63, 37, 56, 36, 72, 37, 57, 61, 70, 50, 77, 40, 80

Offset: 1

Omar E. Pol, Oct 18 2023

nonn

The values of a(n) where a(n) = n give A174973.
The values of a(n) where a(n) is not equal to n give A238524.
If n is an odd prime then a(n) = (n + 1)/2.
Since this is a supersequence of A174973 so all powers of 2 and all even perfect numbers are in the sequence.
Shares infinitely many terms with A365195 from which first differs at a(15).

Row lengths of A365434.

Cf. A000079, A000396, A065091, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238524, A245092, A262626, A347186, A347263, A347367, A365195.

a(2*n-1) >= n.

A378471 Numbers m whose symmetric representation of sigma(m), SRS(m), has at least 2 parts the first of which has width 1.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105

Offset: 1

Hartmut F. W. Hoft, Nov 27 2024

nonn

Numbers m = 2^k * q, k >= 0 and q > 1 odd, without odd prime factors p < 2^(k+1).
This sequence is a proper subsequence of A238524. Numbers 78 = A370206(1) = A238524(55) and 102 = A237287(72) are not in this sequence since their width pattern (A341969) is 1210121.
A000079 is not a subsequence since SRS(2^k), k>=0, consists of a single part of width 1.
Let m = 2^k * q, k >= 0 and q > 1 odd, be a number in this sequence and s the size of the first part of SRS(m) which has width 1 and consists of 2^(k+1) - 1 legs of width 1. Therefore, s = Sum_{i=1..2^(k+1)-1} a237591(m, i) = a235791(m, 1) - a235791(m, 2^(k+1)) = ceiling((m+1)/1 - (1+1)/2) - ceiling((m+1)/2^(k+1) - (2^(k+1) + 1)/2) = (2^(k+1) - 1)(q+1)/2. In other words, point (m, s) is on the line s(m) = (2^(k+1) - 1)/2^(k+1) * m + (2^(k+1) - 1)/2.
For every odd number m in this sequence, the first part of SRS(m) has size (m+1)/2.
Let u = 2^k * Product_{i=1..PrimePi(2^(k+1)} p_i, where p_i is the i-th prime, and let v be the number of elements in this sequence that are in the set V = {m = 2^k * q | 1 < m <= u } then T(j + t*v, k) = T(j, k) + t*u, 1 <= j and 1 <= t, holds for the elements in column k.

			a(5) = 10 is in the sequence since SRS(10) = {9, 9} consists of 2 parts of width 1 and of sizes 9 = (2^2 - 1)(5+1)/2.
a(15) = 25 is in the sequence since the first part of SRS(25) = {13, 5, 13} has width 1 and has size 13 = (2^1 - 1)(25+1)/2.
a(28) = 44 is in the sequence since SRS(44) = {42, 42} has width 1 and has size 42 = (2^3 - 1)(11+1)/2.
The upper left hand 11 X 11 section of array T(j, k) shows the j-th number m in this sequence of the form m = 2^k * q with q odd. The first part of SRS(m) of every number in column k consists of 2^(k+1) - 1 legs of width 1.
j\k| 0   1   2    3    4     5     6      7      8       9       10  ...
------------------------------------------------------------------------
1  | 3   10  44   136  592   2144  8384   32896  133376  527872  2102272
2  | 5   14  52   152  656   2272  8768   33664  133888  528896  2112512
3  | 7   22  68   184  688   2336  8896   34432  138496  531968  2118656
4  | 9   26  76   232  752   2528  9536   34688  140032  537088  2130944
5  | 11  34  92   248  848   2656  9664   35456  142592  538112  2132992
6  | 13  38  116  296  944   2848  10048  35968  144128  543232  2137088
7  | 15  46  124  328  976   3104  10432  36224  145664  544256  2139136
8  | 17  50  148  344  1072  3232  10688  37504  146176  547328  2149376
9  | 19  58  164  376  1136  3296  11072  39296  147712  556544  2161664
10 | 21  62  172  424  1168  3424  11456  39808  150272  558592  2163712
11 | 23  70  188  472  1264  3488  11584  40064  151808  559616  2180096
...
Row 1 is A246956(n), n>=1.
Column 0 is A005408(n) with T(j + 1, 0) = T(j, 0) + 2, n>=1.
Column 1 is A091999(n) with T(j + 2, 1) = T(j, 1) + 12, n>=2.
Column 2 is A270298(n) with T(j + 48, 2) = T(j, 2) + 840, n>=1.
Column 3 is A270301(n) with T(j + 5760, 3) = T(j, 3) + 240240, n>=1.

Cf. A000079, A005408, A091999, A235791, A237287, A237591, A237593, A238524, A246956, A270298, A270301, A341969, A370206, A377654.

(* partsSRS[] and widthPattern[ ] are defined in A377654 *)
a378471[m_, n_] := Select[Range[m, n], Length[partsSRS[#]]>1&&widthPattern[#][[1;;2]]=={1, 0}&]
a378471[1, 105]

cp's OEIS Frontend

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A365195 Height of the first staircase of the ziggurat of order n described in A347186.

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A365433 Height of the ziggurat of order n described in A347186.

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A378471 Numbers m whose symmetric representation of sigma(m), SRS(m), has at least 2 parts the first of which has width 1.