A047836 -id:A047836 - 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.

A174973 Numbers whose divisors increase by a factor of at most 2.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 256

Offset: 1

T. D. Noe, Apr 02 2010

nonn

That is, if the divisors of a number are listed in increasing order, the ratio of adjacent divisors is at most 2. The only odd number in this sequence is 1. Every term appears to be a practical number (A005153). The first practical number not here is 78.
Let p1^e1 * p2^e2 * ... * pr^er be the prime factorization of a number, with primes p1 < p2 < ... < pr and ek > 0. Then the number is in this sequence if and only if pk <= 2*Product_{j < k} p_j^e_j. This condition is similar to, but more restrictive than, the condition for practical numbers.
The polymath8 project led by Terry Tao refers to these numbers as "2-densely divisible". In general they say that n is y-densely divisible if its divisors increase by a factor of y or less, or equivalently, if for every R with 1 <= R <= n, there is a divisor in the interval [R/y,R]. They use this as a weakening of the condition that n be y-smooth. - David S. Metzler, Jul 02 2013
Is this the same as numbers k with the property that the symmetric representation of sigma(k) has only one part? If not, where is the first place these sequences differ? (cf. A237593). - Omar E. Pol, Mar 06 2014
Yes, the sequence so defined is the same as this sequence; see proof in the links. - Hartmut F. W. Hoft, Nov 26 2014
Saias (1997) called these terms "2-dense numbers" and proved that if N(x) is the number of terms not exceeding x, then there are two positive constants c_1 and c_2 such that c_1 * x/log_2(x) <= N(x) <= c_2 * x/log_2(x) for all x >= 2. - Amiram Eldar, Jul 23 2020
Weingartner (2015, 2019) showed that N(x) = c*x/log(x) + O(x/(log(x))^2), where c = 1.224830... As a result, a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 0.816439... - Andreas Weingartner, Jun 22 2021

			The divisors of 12 are 1, 2, 3, 4, 6, 12. The ratios of adjacent divisors is 2, 3/2, 4/3, 3/2, and 2, which are all <= 2. Hence 12 is in this sequence.
Example from _Omar E. Pol_, Mar 06 2014: (Start)
    The symmetric representation of sigma(6) = 12 in the first quadrant looks like this:
   y
   .
   ._ _ _ _
   |_ _ _  |_
   .     |   |_
   .     |_ _  |
   .         | |
   .         | |
   . . . . . |_| . . x
.
6 is in the sequence because the symmetric representation of sigma(6) = 12 has only one part. The 6th row of A237593 is [4, 1, 1, 1, 1, 4] and the 5th row of A237593 is [3, 2, 2, 3] therefore between both symmetric Dyck paths there is only one region (or part) of size 12.
    70 is not in the sequence because the symmetric representation of sigma(70) = 144 has three parts. The 70th row of A237593 is [36, 12, 6, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 6, 12, 36] and the 69th row of A237593 is [35, 12, 7, 4, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 7, 12, 35] therefore between both symmetric Dyck paths there are three regions (or parts) of size [54, 36, 54]. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
José Manuel Rodríguez Caballero, Jordan's Expansion of the Reciprocal of Theta Functions and 2-densely Divisible Numbers, Integers, Vol. 20 (2020), Article A2.
Hartmut F. W. Hoft, Proof of a conjecture.
Hartmut F. W. Hoft, Proof of a second conjecture.
Eric Saias, Entiers à diviseurs denses 1, Journal of Number Theory, Vol. 62, No. 1 (1997), pp. 163-191.
Terence Tao, A Truncated Elementary Selberg Sieve of Pintz. (blog entry defining y-densely divisible)
Terence Tao et al., Polymath8 home page.
Andreas Weingartner, Practical numbers and the distribution of divisors, The Quarterly Journal of Mathematics, Vol. 66, No. 2 (2015), pp. 743-758; arXiv preprint, arXiv:1405.2585 [math.NT], 2014-2015.
Andreas Weingartner, On the constant factor in several related asymptotic estimates, Math. Comp., Vol. 88, No. 318 (2019), pp. 1883-1902; arXiv preprint, arXiv:1705.06349 [math.NT], 2017-2018.
Andreas Weingartner, The number of prime factors of integers with dense divisors, arXiv:2101.11585 [math.NT], 2021.
Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, arXiv:2303.16819 [math.NT], 2023.

Subsequence of A196149 and of A071562. A000396 and A000079 are subsequences.
Cf. A027750, A047836, A237593, A365429 (characteristic function).
Column 1 of A240062.
First differs from A103288 and A125225 at a(23). First differs from A005153 at a(24).

a174973 n = a174973_list !! (n-1)
a174973_list = filter f [1..] where
   f n = all (<= 0) $ zipWith (-) (tail divs) (map (* 2) divs)
         where divs = a027750_row' n
-- Reinhard Zumkeller, Jun 25 2015, Sep 28 2011

[k:k in [1..260]|forall{i:i in [1..#Divisors(k)-1]|d[i+1]/d[i] le 2 where d is Divisors(k)}]; // Marius A. Burtea, Jan 09 2020

a:= proc() option remember; local k; for k from 1+`if`(n=1, 0,
      a(n-1)) while (l-> ormap(x-> x, [seq(l[i]>l[i-1]*2, i=2..
      nops(l))]))(sort([(numtheory[divisors](k))[]])) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 27 2018

OK[n_] := Module[{d=Divisors[n]}, And@@(#<=2& /@ (Rest[d]/Most[d]))]; Select[Range[1000], OK]
dif2Q[n_]:=AllTrue[#[[2]]/#[[1]]&/@Partition[Divisors[n],2,1],#<=2&]; Select[Range[300],dif2Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 29 2020 *)

is(n)=my(d=divisors(n));for(i=2,#d,if(d[i]>2*d[i-1],return(0)));1 \\ Charles R Greathouse IV, Jul 06 2013

from sympy import divisors
def ok(n):
    d = divisors(n)
    return all(d[i]/d[i-1] <= 2 for i in range(1, len(d)))
print(list(filter(ok, range(1, 257)))) # Michael S. Branicky, Jun 22 2021

a(n) = A047836(n) / 2. - Reinhard Zumkeller, Sep 28 2011

a(n) = C*n*log(n*log(n)) + O(n), where C = 0.816439... (see comments). - Andreas Weingartner, Jun 23 2021

Edited by N. J. A. Sloane, Sep 09 2023

Edited by Peter Munn, Oct 17 2023

A047988 Start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. a(n) is the smallest cost to reach 2.

Original entry on oeis.org

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

Offset: 2

Thomas Kantke (bytes.more(AT)ibm.net)

			From _David A. Corneth_, Jun 05 2020: (Start)
For n = 19 we get to 2 via
n := 19 - 1 = 2 (+1 point as we subtract 1 from n)
n := 18/3 = 6 (+0 point as we divide by a divisor)
n := 6/2 = 3 (+0 point, we divide by a divisor. we can't divide by 3 as 3 > sqrt(6)).
n := 3 - 1 = 2 (+1 point; steps end here; we have n = 2)
We earned 2 points in these steps. As this is the minimal number of points starting at n = 19, a(19) = 2. (End)

David A. Corneth, Table of n, a(n) for n = 2..10001
David A. Corneth, PARI program
Thomas Kantke, Das Spiel Minimum und die Zerlegung natürlicher Zahlen, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.

Cf. A047836, A047984, A047985, A047986, A047987.

Cf. A048823, A048824, A048825, A048826, A048827.

\\ see Corneth link \\ David A. Corneth, Jun 05 2020

More terms from David W. Wilson

A047984 Integers n such that A047988(n)=1.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 37, 39, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 57, 59, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 88, 90, 91, 92, 93, 94, 95, 97

Offset: 1

Thomas Kantke (bytes.more(AT)ibm.net)

nonn

			7 has value 1 since we add 1 to get 8, then divide by 2 twice.

David A. Corneth, Table of n, a(n) for n = 1..10000
Thomas Kantke, Das Spiel Minimum und die Zerlegung natürlicher Zahlen, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.

Cf. A047836, A047985, A047986, A047987, A047988.

More terms from David W. Wilson

A047987 Integers n such that A047988(n)=4.

Original entry on oeis.org

3976733, 8053483, 9942523, 10197427, 15126557, 28623773, 32269037, 34438387, 36627413, 39593803, 47812747, 49120843, 55066867, 69909667, 78068443, 80024213, 83402597, 89512483, 97799033, 101652883, 106675637, 116621213, 120317683, 128241067

Offset: 1

Thomas Kantke (bytes.more(AT)ibm.net)

nonn

Thomas Kantke, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.

Cf. A047836, A047984-A047988.

More terms from Max Alekseyev, Jun 13 2011

A047985 Integers n such that A047988(n)=2.

Original entry on oeis.org

19, 29, 38, 43, 53, 58, 67, 76, 86, 87, 89, 101, 103, 106, 116, 134, 137, 139, 149, 151, 152, 157, 163, 172, 174, 178, 197, 202, 203, 206, 211, 212, 227, 229, 232, 233, 247, 261, 267, 268, 269, 271, 274, 277, 278, 283, 293, 298, 302, 303, 304, 307

Offset: 1

Thomas Kantke (bytes.more(AT)ibm.net)

nonn

			19 has value 2 since by adding 1 we reach 20 which has value 1; there is no cheaper way to reach 2.

David A. Corneth, Table of n, a(n) for n = 1..10000
Thomas Kantke, Das Spiel Minimum und die Zerlegung natürlicher Zahlen, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.

Cf. A047836, A047984, A047986, A047987, A047988.

More terms from David W. Wilson

A047986 Integers n such that A047988(n)=3.

Original entry on oeis.org

173, 347, 823, 907, 1237, 1697, 2137, 2333, 2423, 2473, 2767, 2777, 3253, 3413, 3547, 3559, 3623, 3767, 4243, 4273, 4457, 4547, 4723, 5091, 5297, 5861, 6185, 6197, 6593, 6823, 6827, 6871, 7093, 7247, 7537, 7717, 7817, 7823, 7879, 8243

Offset: 1

Thomas Kantke (bytes.more(AT)ibm.net)

nonn

David A. Corneth, Table of n, a(n) for n = 1..10000
Thomas Kantke, Das Spiel Minimum und die Zerlegung natürlicher Zahlen, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.

Cf. A047836, A047985, A047986, A047987, A047988.

More terms from David W. Wilson

A317305 Sum of divisors of the n-th number whose divisors increase by a factor of 2 or less.

Original entry on oeis.org

1, 3, 7, 12, 15, 28, 31, 39, 42, 60, 56, 72, 63, 91, 90, 96, 124, 120, 120, 168, 127, 144, 195, 186, 224, 180, 234, 252, 217, 210, 280, 248, 360, 312, 255, 336, 336, 403, 372, 392, 378, 363, 480, 372, 546, 508, 399, 468, 465, 504, 434, 576, 600, 504, 504, 560, 546, 744, 728, 511

Offset: 1

Omar E. Pol, Aug 25 2018

nonn

Also consider the n-th number k with the property that the symmetric representation of sigma(k) has only one part. a(n) is the area of the diagram (see the example). For more information see A237593 and its related sequences.

			Illustration of initial terms (n = 1..13):
.
  a(n)
        _ _   _   _   _       _       _   _   _       _       _   _   _
   1   |_| | | | | | | |     | |     | | | | | |     | |     | | | | | |
   3   |_ _|_| | | | | |     | |     | | | | | |     | |     | | | | | |
        _ _|  _|_| | | |     | |     | | | | | |     | |     | | | | | |
   7   |_ _ _|    _|_| |     | |     | | | | | |     | |     | | | | | |
        _ _ _|  _|  _ _|     | |     | | | | | |     | |     | | | | | |
  12   |_ _ _ _|  _|    _ _ _| |     | | | | | |     | |     | | | | | |
        _ _ _ _| |    _|    _ _|     | | | | | |     | |     | | | | | |
  15   |_ _ _ _ _|  _|     |    _ _ _| | | | | |     | |     | | | | | |
                   |      _|   |  _ _ _|_| | | |     | |     | | | | | |
                   |  _ _|    _| |    _ _ _|_| |     | |     | | | | | |
        _ _ _ _ _ _| |      _|  _|   |  _ _ _ _|     | |     | | | | | |
  28   |_ _ _ _ _ _ _|  _ _|  _|  _ _| |    _ _ _ _ _| |     | | | | | |
                       |  _ _|  _|    _|   |    _ _ _ _|     | | | | | |
                       | |     |     |  _ _|   |    _ _ _ _ _| | | | | |
        _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |   |  _ _ _ _ _|_| | | |
  31   |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|   | |    _ _ _ _ _|_| |
        _ _ _ _ _ _ _ _ _| | |     |      _|    _ _| |   |  _ _ _ _ _ _|
  39   |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|  _ _|  _ _|   | |
        _ _ _ _ _ _ _ _ _ _| | |       |   |    _|    _ _| |
  42   |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|  _|     |  _ _|
                               | |       |  _|      _| |
                               | |  _ _ _| |      _|  _|
        _ _ _ _ _ _ _ _ _ _ _ _| | |  _ _ _|  _ _|  _|
  60   |_ _ _ _ _ _ _ _ _ _ _ _ _| | |       |  _ _|
                                   | |  _ _ _| |
                                   | | |  _ _ _|
        _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | |
  56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
  72   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
        _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  63   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The length of the largest Dyck path of the n-th diagram equals A047836(n).
The semilength equals A174973(n).
a(n) is the area of the n-th diagram.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

A317307 is a subsequence.
Cf. A174973.
Cf. A000203, A047836, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A262626, A361208.

A317305[upto_]:=Table[If[AllTrue[Map[Last[#]/First[#]&,Partition[Divisors[n],2,1]],#<=2&],DivisorSigma[1,n],Nothing],{n,upto}];
A317305[500] (* Paolo Xausa, Jan 12 2023 *)

a(n) = A000203(A174973(n)).

A212524 Number of k < 10^n such that A047988(k) = 0.

Original entry on oeis.org

3, 17, 108, 755, 5936, 48474, 406270, 3532031, 31295358, 279591668, 2521429242, 22996137423

Offset: 1

Martin Renner, Jun 13 2012

nonn

Thomas Kantke, Mathematische Unterhaltungen, in: Spektrum der Wissenschaft 4/1993, p. 11-13.
Thomas Kantke, Das Spiel Minimum, in: Spektrum der Wissenschaft Spezial Physik - Mathematik - Technik 2/2012, pp. 57-66.

Cf. A047836, A047988, A213530, A213531, A213532, A213533.

A052287 Start with 3; the general rule is "if x is present then so is x*y for every y <= x".

Original entry on oeis.org

3, 6, 9, 12, 18, 24, 27, 30, 36, 45, 48, 54, 60, 63, 72, 81, 84, 90, 96, 108, 120, 126, 132, 135, 144, 150, 162, 168, 180, 189, 192, 198, 210, 216, 225, 234, 240, 243, 252, 264, 270, 288, 297, 300, 306, 312, 315, 324, 330, 336, 351, 360, 378, 384, 390, 396

Offset: 1

Giuseppe Melfi, Feb 08 2000

			63 is an element because 63 = 3*3*7 and 3 <= 3 and 7 <= 3*3.

T. D. Noe, Table of n, a(n) for n = 1..1000

If instead we start with 2, we obtain the "Nullwertzahlen sequence" A047836.

Cf. A196149.

import Data.List.Ordered (union)
a052287 n = a052287_list !! (n-1)
a052287_list = f [3] where
   f (x:xs) = x : f (xs `union` map (x *) [2..x])
-- Reinhard Zumkeller, Jun 25 2015, Sep 28 2011

N:= 1000: # get all terms <= N
S:= {3}:
New:= {3}:
while New <> {} do
  x:= New[1];
  New:= subsop(1=NULL,New);
  R:= {seq(k*x, k=1..min(x,N/x))} minus S;
  S:= S union R;
  New:= New union R;
od:
sort(convert(S,list));  # Robert Israel, Aug 27 2015

3 Select[Range[132], Max[#[[2]]/#[[1]] & /@ Partition[Divisors[#], 2, 1]] <= 3 &] (* Michael De Vlieger, Aug 27 2015, after Harvey P. Dale at A196149 *)

x is a term if and only if x = 3*p1*p2*...*pk with primes 2 <= p1 <= p2 <= ... <= pk and 3*p1*p2*...*pi >= p(i+1) for all i < k.
a(n) = 3 * A196149(n). - Reinhard Zumkeller, Sep 28 2011
The number of terms <= x is c*x/log(x) + O(x/(log(x))^2), where c = 0.68514..., and a(n) = C*n*log(n*log(n)) + O(n), where C = 1/c = 1.45954... This follows from the formula just above. - Andreas Weingartner, Jun 30 2021

More terms from Reinhard Zumkeller, Jun 22 2003

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.

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A174973 Numbers whose divisors increase by a factor of at most 2.

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A047988 Start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. a(n) is the smallest cost to reach 2.

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A047984 Integers n such that A047988(n)=1.

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A047987 Integers n such that A047988(n)=4.

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A047985 Integers n such that A047988(n)=2.

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A047986 Integers n such that A047988(n)=3.

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A317305 Sum of divisors of the n-th number whose divisors increase by a factor of 2 or less.

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A212524 Number of k < 10^n such that A047988(k) = 0.

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