A005279 -id:A005279 - cp's OEIS frontend

A020886 Ordered semiperimeters of primitive Pythagorean triangles.

6, 15, 20, 28, 35, 42, 45, 63, 66, 72, 77, 88, 91, 99, 104, 110, 117, 120, 130, 143, 153, 156, 165, 170, 187, 190, 195, 204, 209, 210, 221, 228, 231, 238, 247, 255, 266, 272, 273, 276, 285, 299, 304, 322, 323, 325, 336, 342, 345, 350, 357, 368, 378, 391, 399

Offset: 1

Clark Kimberling

nonn

k is in this sequence iff A078926(k) > 0.
Also, ordered sides c of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c (A343893). - Bernard Schott, May 06 2021
a(n) are the ordered radii of inscribed circles in squares, from which the tangents to the circles are cut off by primitive Pythagorean triangles. - Alexander M. Domashenko, Oct 17 2024

Ray Chandler, Table of n, a(n) for n = 1..10000 (duplicates removed by Sean A. Irvine)

Subsequence of A005279.
Cf. A024364, A078926, A020882, A020884, A020885.
Triangles with 2/a = 1/b + 1/c:  A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

isA020886 := proc(an) local r::integer,s::integer ; for r from floor((an/2)^(1/2)) to floor(an^(1/2)) do for s from r-1 to 1 by -2 do if r*(r+s) = an and gcd(r,s) < 2 then RETURN(true) ; fi ; if r*(r+s) < an then break ; fi ; od ; od : RETURN(false) ; end : for n from 2 to 400 do if isA020886(n) then printf("%d,",n) ; fi ; od ; # R. J. Mathar, Jun 08 2006

A078926[n_] := Sum[Boole[n < d^2 < 2n && CoprimeQ[d, n/d]], {d, Divisors[ n/2^IntegerExponent[n, 2]]}];
Select[Range[1000], A078926[#]>0&] (* Jean-François Alcover, Mar 23 2020 *)

is(n,f=factor(n))=my(P=apply(i->f[i,1]^f[i,2],[2-n%2..#f~]),nn=2*n); forvec(v=vector(#P,i,[0,1]), my(d=prod(i=1,#v,P[i]^v[i]),d2=d^2); if(d2n, return(1))); 0
list(lim)=my(v=List()); forfactored(n=6,lim\1, if(is(n[1],n[2]), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Feb 03 2023

a(n) = A024364(n)/2.

A239657 Number of odd divisors m of n such that there is a divisor d of n with d < m < 2*d.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 3, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 1, 0, 0, 3, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 1, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 5, 1, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 2, 0, 1, 4, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 0, 5, 0, 0

Offset: 1

Omar E. Pol, Mar 23 2014

nonn

The original name was: Number of odd divisors of n minus the number of parts in the symmetric representation of sigma(n).
Observation: at least the indices of the first 42 positive elements coincide with A005279: 6, 12, 15, 18, 20, 24..., checked (by hand) up to n = 2^7.
The observation is true for the indices of all positive elements. Hence the indices of the zeros give A174905. - Omar E. Pol, Jan 06 2017
a(n) is the number of subparts minus the number of parts in the symmetric representation of sigma(n). For the definition of "subpart" see A279387. - Omar E. Pol, Sep 26 2018
a(n) is the number of subparts of the symmetric representation of sigma(n) that are not in the first layer. - Omar E. Pol, Jan 26 2025

			Illustration of the symmetric representation of sigma(15) = 24 in the third quadrant:
.      _
.     | |
.     | |
.     | |
.     | |
.     | |
.     | |
.     | |
.     |_|_ _ _
.    8      | |_ _
.           |_    |
.             |_  |_
.            8  |_ _|
.                   |
.                   |_ _ _ _ _ _ _ _
.                   |_ _ _ _ _ _ _ _|
.                 8
.
For n = 15 the divisors of 15 are 1, 3, 5, 15, so the number of odd divisors of 15 is equal to 4. On the other hand the parts of the symmetric representation of sigma(15) are [8, 8, 8], there are three parts, so a(15) = 4 - 3 = 1.
From _Omar E. Pol_, Sep 26 2018: (Start)
Also the number of odd divisors of 15 equals the number of partitions of 15 into consecutive parts and equals the number of subparts in the symmetric representation of sigma(15). Then we have that the number of subparts minus the number of parts is  4 - 3 = 1, so a(15) = 1.
.      _
.     | |
.     | |
.     | |
.     | |
.     | |
.     | |
.     | |
.     |_|_ _ _
.    8      | |_ _
.           |_ _  |
.          7  |_| |_
.            1  |_ _|
.                   |
.                   |_ _ _ _ _ _ _ _
.                   |_ _ _ _ _ _ _ _|
.                 8
.
The above diagram shows the symmetric representation of sigma(15) with its four subparts: [8, 7, 1, 8]. (End)
From _Omar E. Pol_, Mar 30 2025: (Start)
The above diagram also shows that in the first layer there are three parts (having sizes [8, 7, 8]). Also there is another part that is not in the first layer, so a(15) = 1.
On the other hand for n = 15 there is only one odd divisor m of 15 such that  d < m < 2*d and d divides 15. That odd divisor is 5 as shown below, so a(15) = 1.
   d  <  m  <  2*d
--------------------
   1            2
   3     5      6
   5           10
  15           30
.
For n = 18 there are two odd divisors m of 18 such that  d < m < 2*d and d divides 18. Those odd divisors are 3 and 9 as shown below, so a(18) = 2.
   d  <  m  <  2*d
--------------------
   1            2
   2     3      4
   3            6
   6     9     12
   9           18
  18           36
.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..5000 (computed from the b-file of A237271 provided by Michel Marcus)

Cf. A000203, A001227, A005279, A174905, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A279387, A279391, A262626, A286000, A286001, A296508, A347186, A383147.

a[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, Count[d, ?(OddQ[#[[2]]] && #[[2]] < 2*#[[1]] &)]]; Array[a, 100] (* _Amiram Eldar, Apr 01 2025 *)

a(n) = A001227(n) - A237271(n).

New Name from Omar E. Pol, Jan 26 2025

A174905 Numbers with no pair (d,e) of divisors such that d < e < 2*d.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 76, 79, 81, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 98, 101, 103, 106

Offset: 1

Reinhard Zumkeller, Apr 01 2010

nonn

A174903(a(n)) = 0; complement of A005279;
sequences of powers of primes are subsequences;
a(n) = A129511(n) for n < 27, A129511(27) = 35 whereas a(27) = 37.
Also the union of A241008 and A241010 (see the link for a proof). - Hartmut F. W. Hoft, Jul 02 2015
In other words: numbers n with the property that all parts in the symmetric representation of sigma(n) have width 1. - Omar E. Pol, Dec 08 2016
Sequence A357581 shows the numbers organized in columns of a square array by the number of parts in their symmetric representation of sigma. - Hartmut F. W. Hoft, Oct 04 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Hartmut F. W. Hoft, Proof that this sequence equals union of A241008 and A241010

Cf. A000040, A000961, A001248, A005279, A030078, A030514, A129511, A174903, A237271, A237593, A241008, A241010.

Cf. A357581.

a174905 n = a174905_list !! (n-1)
a174905_list = filter ((== 0) . a174903) [1..]
-- Reinhard Zumkeller, Sep 29 2014

filter:= proc(n)
  local d,q;
   d:= numtheory:-divisors(n);
   min(seq(d[i+1]/d[i],i=1..nops(d)-1)) >= 2
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 08 2014

(* it suffices to test adjacent divisors *)
a174905[n_] := Module[{d = Divisors[n]}, ! Apply[Or, Map[2 #[[1]] > #[[2]] &, Transpose[{Drop[d, -1], Drop[d, 1]}]]]]
(* Hartmut F. W. Hoft, Aug 07 2014 *)
Select[Range[106], !MatchQ[Divisors[#], {_, d_, e_, _} /; e < 2d]& ] (* Jean-François Alcover, Jan 31 2018 *)

A279391 Irregular triangle read by rows in which row n lists the subparts of the successive layers of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 3, 2, 2, 7, 3, 3, 11, 1, 4, 4, 15, 5, 3, 5, 9, 9, 6, 6, 23, 5, 7, 7, 12, 12, 8, 7, 8, 1, 31, 9, 9, 35, 2, 2, 10, 10, 39, 3, 11, 5, 5, 11, 18, 18, 12, 12, 47, 13, 13, 5, 13, 21, 21, 14, 6, 6, 14, 55, 1, 15, 15, 59, 3, 7, 3, 16, 16, 63, 17, 7, 7, 17, 27, 27, 18, 9, 18, 3, 71, 10, 10, 19, 19, 30, 30

Offset: 1

Omar E. Pol, Dec 12 2016

Note that the terms in the n-th row are the same as the terms in the n-th row of triangle A280851, but in some rows the terms appear in distinct order. First differs from A280851 at a(28) = T(15,3). - Omar E. Pol, Apr 24 2018

Row n in the triangle is a sequence of A250068(n) symmetric sections, each section consisting of the sizes of the subparts on that level in the symmetric representation of sigma of n - from the top down in the images below or left to right as drawn in A237593. - Hartmut F. W. Hoft, Sep 05 2021

			Triangle begins (first 15 rows):
   [1];
   [3];
   [2, 2];
   [7];
   [3, 3];
   [11], [1];
   [4, 4];
   [15];
   [5, 3, 5];
   [9, 9];
   [6, 6];
   [23], [5];
   [7, 7];
   [12, 12];
   [8, 7, 8], [1];
  ...
For n = 12 we have that the 11th row of triangle A237593 is [6, 3, 1, 1, 1, 1, 3, 6] and the 12th row of the same triangle is [7, 2, 2, 1, 1, 2, 2, 7], so the diagram of the symmetric representation of sigma(12) = 28 is constructed as shown below in Figure 1:
.                          _                                    _
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                    _ _ _| |                             _ _ _| |
.                  _|    _ _|                           _|  _ _ _|
.                _|     |                             _|  _| |
.               |      _|                            |  _|  _|
.               |  _ _|                              | |_ _|
.    _ _ _ _ _ _| |    28                 _ _ _ _ _ _| |    5
.   |_ _ _ _ _ _ _|                      |_ _ _ _ _ _ _|
.                                                       23
.
.   Figure 1. The symmetric            Figure 2. After the dissection
.   representation of sigma(12)        of the symmetric representation
.   has only one part which            of sigma(12) into layers of
.   contains 28 cells, so              width 1 we can see two "subparts"
.   the 12th row of the                that contain 23 and 5 cells
.   triangle A237270 is [28].          respectively, so the 12th row of
.                                      this triangle is [23], [5].
.
For n = 15 we have that the 14th row of triangle A237593 is [8, 3, 1, 2, 2, 1, 3, 8] and the 15th row of the same triangle is [8, 3, 2, 1, 1, 1, 1, 2, 3, 8], so the diagram of the symmetric representation of sigma(15) = 24 is constructed as shown below in Figure 3:
.                                _                                  _
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                          _ _ _|_|                           _ _ _|_|
.                      _ _| |      8                      _ _| |      8
.                     |    _|                            |  _ _|
.                    _|  _|                             _| |_|
.                   |_ _|  8                           |_ _|  1
.                   |                                  |    7
.    _ _ _ _ _ _ _ _|                   _ _ _ _ _ _ _ _|
.   |_ _ _ _ _ _ _ _|                  |_ _ _ _ _ _ _ _|
.                    8                                  8
.
.   Figure 3. The symmetric            Figure 4. After the dissection
.   representation of sigma(15)        of the symmetric representation
.   has three parts of size 8          of sigma(15) into layers of
.   because every part contains        width 1 we can see four "subparts".
.   8 cells, so the 15th row of        The first layer has three subparts:
.   triangle A237270 is [8, 8, 8].     8, 7, 8. The second layer has
.                                      only one subpart of size 1, so
.                                      the 15th row of this triangle is
.                                      [8, 7, 8], [1].
.
The smallest even number with 3 levels is 60; its row of subparts is: [119], [37], [6, 6]. The smallest odd number with 3 levels is 315; its row of subparts is:  [158, 207, 158], [11, 26, 5, 9, 5, 26, 11], [4, 4]. - _Hartmut F. W. Hoft_, Sep 05 2021

Index entries for sequences related to sigma(n)

The length of row n equals A001227(n).
If n is odd the length of row n equals A000005(n).
Row sums give A000203.
For the definition of "subparts" see A279387.
For the triangle of sums of subparts see A279388.
Cf. A001227, A005279, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239657, A244050, A245092, A250068, A250070, A261699, A262626, A280851, A296508.

(* support functions are defined in aA237593 and A262045 *)
subP[level_] := Module[{s=Map[Apply[Plus, #]&, Select[level, First[#]!=0&]]}, If[OddQ[Length[s]], s[[(Length[s]+1)/2]]-=1]; s]
a279391[n_] := Module[{widL=a262045[n], lenL=a237593[n], srs, subs}, srs=Transpose[Map[PadRight[If[widL[[#]]>0, Table[1, widL[[#]]], {0}], Max[widL]]&, Range[Length[lenL]]]]; subs=Map[SplitBy[lenL srs[[#]], #!=0&]&, Range[Max[widL]]]; Flatten[Map[subP, subs]]]
Flatten[Map[a279391, Range[38]]] (* Hartmut F. W. Hoft, Sep 05 2021 *)

A379288 Irregular triangle read by rows in which row n lists the odd divisors of n excluding odd divisors e for which there exists another divisor j with j < e < 2*j.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 5, 1, 1, 7, 1, 1, 3, 9, 1, 5, 1, 11, 1, 1, 13, 1, 7, 1, 3, 15, 1, 1, 17, 1, 1, 19, 1, 1, 3, 7, 21, 1, 11, 1, 23, 1, 1, 5, 25, 1, 13, 1, 3, 9, 27, 1, 1, 29, 1, 1, 31, 1, 1, 3, 11, 33, 1, 17, 1, 5, 35, 1, 1, 37, 1, 19, 1, 3, 13, 39, 1, 1, 41, 1, 1, 43

Offset: 1

Omar E. Pol, Dec 21 2024

The excluded divisors are the odd divisors e listed in A005279.
Conjecture 1: the row lengths are given by A237271 (true for at least the first 10000 terms of A237271)
From Hartmut F. W. Hoft, Jan 09 2025: (Start)
Proof of Conjecture 1:
An entire part of SRS(n), n = 2^k * q with k >= 0 and q odd, up to the diagonal is described in row n of A249223 by a 1 in position d, an odd divisor of n, 0's in positions d-1 and 2^(k+1) * f, f >= d an odd divisor of n, and nonzero numbers that increase or decrease by 1 in between.
The odd divisors e of n with d < e < 2^(k+1) * f are the "e" odd divisors of A005279 since for divisor s of n, d < s < e < 2*s < 2^(k+1) * f holds.
The odd divisors u of n greater than A003056(n) are encoded by the 2^(k+1) * f above as u = q/f and odd divisors d < A003056(n) are also encoded as 2^(k+1) * q/d. Then odd divisors e of n with q/f < e < 2^(k+1) * q/d are the "e" odd divisors of A005279 since for divisor t of n,  q/f < t < e < 2*t < 2^(k+1) * q/d holds.
For a part containing the diagonal the inequalities above hold on the respective sides of the diagonal.
As a consequence the number of entries in row n of this triangle equals A237271(n). (End)
From Omar E. Pol, Jun 26 2025: (Start)
Conjecture 2: T(n,m) is the smallest number in the m-th 2-dense sublist of divisors of n.
We call "2-dense sublists of divisors of n" to the maximal sublists of divisors of n whose terms increase by a factor of at most 2.
In a 2-dense sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.
If the conjecture is true so row sums give A379379 and the row lengths give A237271, and the same row lengths have the sequences A384222, A384225 and A384226. Also the conjecture of A384149 should be true.
Observation: at least for the first 5000 rows (the first 15542 terms) the conjecture 2  coincides with the definition from the Name section and the row lengths give A237271.
An example of the conjecture 2, for n = 1..24 is as shown below:
  -------------------------------------------------------------------
  |  n |  Row n of        |  List of divisors of n       | Number of |
  |    |  the triangle    |  [with sublists in brackets] | sublists  |
  --------------------------------------------------------------------
  |  1 |   1;             |  [1];                        |     1     |
  |  2 |   1;             |  [1, 2];                     |     1     |
  |  3 |   1,  3;         |  [1], [3];                   |     2     |
  |  4 |   1;             |  [1, 2, 4];                  |     1     |
  |  5 |   1,  5;         |  [1], [5];                   |     2     |
  |  6 |   1;             |  [1, 2, 3, 6];               |     1     |
  |  7 |   1,  7;         |  [1], [7];                   |     2     |
  |  8 |   1;             |  [1, 2, 4, 8];               |     1     |
  |  9 |   1,  3,  9;     |  [1], [3], [9];              |     3     |
  | 10 |   1,  5;         |  [1, 2], [5, 10];            |     2     |
  | 11 |   1, 11;         |  [1], [11];                  |     2     |
  | 12 |   1;             |  [1, 2, 3, 4, 6, 12];        |     1     |
  | 13 |   1, 13;         |  [1], [13];                  |     2     |
  | 14 |   1,  7;         |  [1, 2], [7, 14];            |     2     |
  | 15 |   1,  3, 15;     |  [1], [3, 5], [15];          |     3     |
  | 16 |   1;             |  [1, 2, 4, 8, 16];           |     1     |
  | 17 |   1, 17;         |  [1], [17];                  |     2     |
  | 18 |   1;             |  [1, 2, 3, 6, 9, 18];        |     1     |
  | 19 |   1, 19;         |  [1], [19];                  |     2     |
  | 20 |   1;             |  [1, 2, 4, 5, 10, 20];       |     1     |
  | 21 |   1,  3,  7, 21; |  [1], [3], [7], [21];        |     4     |
  | 22 |   1, 11;         |  [1, 2], [11, 22];           |     2     |
  | 23 |   1, 23;         |  [1], [23];                  |     2     |
  | 24 |   1;             |  [1, 2, 3, 4, 6, 8, 12, 24]; |     1     |
   ...
For n = 10 the list of divisors of 10 is [1, 2, 5, 10]. There are two 2-dense sublists of divisors of 10, they are [1, 2], [5, 10]. The smallest numbers in the sublists are [1, 5] respectively, so the row 10 is [1, 5].
For n = 15 the list of divisors of 15 is [1, 3, 5, 15]. There are three 2-dense sublists of divisors of 15, they are [1], [3, 5], [15]. The smallest numbers in the sublists are [1, 3, 15] respectively, so the row 15 is [1, 3, 15].
78 is the first practical number A005153 not in A174973. For n = 78 the list of divisors of 78 is [1, 2, 3, 6, 13, 26, 39, 78]. There are two 2-dense sublists of divisors of 78, they are [1, 2, 3, 6] and [13, 26, 39, 78]. The smallest numbers in the sublists are [1, 13] respectively, so the row 78 is [1, 13].
(End)
Conjecture 3: T(n,m) is the m-th divisor p of n such that p is greater than twice the adjacent previous divisor of n. - Omar E. Pol, Aug 02 2025

These are the odd terms of A379374.
Subsequence of A182469.
Row sums give A379379.
Cf. A001227, A005279, A237270, A237271, A237593, A239657, A379384, A379461, A379634.
Cf. A000203, A003056, A249223.
Cf. A005153, A174973, A384149, A384222, A384225, A384226, A385000.

row[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, Select[Join[{1}, Select[d, #[[2]] >= 2*#[[1]] &][[;; , 2]]], OddQ]]; Table[row[n], {n, 1, 50}] // Flatten (* Amiram Eldar, Dec 22 2024 *)

More terms from Amiram Eldar, Dec 22 2024

A279388 Irregular triangle read by rows: T(n,k) is the sum of the subparts in the k-th layer of the symmetric representation of sigma(n), if such a layer exists.

Original entry on oeis.org

1, 3, 4, 7, 6, 11, 1, 8, 15, 13, 18, 12, 23, 5, 14, 24, 23, 1, 31, 18, 35, 4, 20, 39, 3, 32, 36, 24, 47, 13, 31, 42, 40, 55, 1, 30, 59, 13, 32, 63, 48, 54, 45, 3, 71, 20, 38, 60, 56, 79, 11, 42, 83, 13, 44, 84, 73, 5, 72, 48, 95, 29, 57, 93, 72, 98, 54, 107, 13, 72, 111, 9, 80, 90, 60, 119, 37, 12

Offset: 1

Omar E. Pol, Dec 12 2016

			Triangle begins (first 15 rows):
  1;
  3;
  4;
  7;
  6;
  11, 1;
  8;
  15;
  13;
  18;
  12;
  23, 5;
  14;
  24;
  23, 1;
  ...
For n = 12 we have that the 11th row of triangle A237593 is [6, 3, 1, 1, 1, 1, 3, 6] and the 12th row of the same triangle is [7, 2, 2, 1, 1, 2, 2, 7], so the diagram of the symmetric representation of sigma(12) = 28 is constructed as shown below in Figure 1:
.                          _                                    _
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                         | |                                  | |
.                    _ _ _| |                             _ _ _| |
.              28  _|    _ _|                       23  _|  _ _ _|
.                _|     |                             _|  _| |
.               |      _|                            |  _|  _|
.               |  _ _|                              | |_ _|
.    _ _ _ _ _ _| |                       _ _ _ _ _ _| |      5
.   |_ _ _ _ _ _ _|                      |_ _ _ _ _ _ _|
.
.   Figure 1. The symmetric            Figure 2. After the dissection
.   representation of sigma(12)        of the symmetric representation
.   has only one part which            of sigma(12) into layers of
.   contains 28 cells, so              width 1 we can see two "subparts"
.   A000203(12) = 28.                  that contain 23 and 5 cells
.                                      respectively, so the 12th row of
.                                      this triangle is [23, 5].
.
For n = 15 we have that the 14th row of triangle A237593 is [8, 3, 1, 2, 2, 1, 3, 8] and the 15th row of the same triangle is [8, 3, 2, 1, 1, 1, 1, 2, 3, 8], so the diagram of the symmetric representation of sigma(15) is constructed as shown below in Figure 3:
.                                _                                  _
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                               | |                                | |
.                           8   | |                            8   | |
.                               | |                                | |
.                               | |                                | |
.                          _ _ _|_|                           _ _ _|_|
.                   8  _ _| |                          7  _ _| |
.                     |    _|                            |  _ _|
.                    _|  _|                             _| |_|
.                   |_ _|                              |_ _|  1
.           8       |                          8       |
.    _ _ _ _ _ _ _ _|                   _ _ _ _ _ _ _ _|
.   |_ _ _ _ _ _ _ _|                  |_ _ _ _ _ _ _ _|
.
.   Figure 3. The symmetric            Figure 4. After the dissection
.   representation of sigma(15)        of the symmetric representation
.   has three parts of size 8,         of sigma(15) into layers of
.   whose sum is 8 + 8 + 8 = 24,       width 1 we can see four "subparts".
.   so A000203(15) = 24.               The first layer has three subparts
.                                      whose sum is 8 + 7 + 8 = 23. The
.                                      second layer has only one subpart
.                                      of size 1, so the 15th row of this
.                                      triangle is [23, 1].
.

Index entries for sequences related to sigma(n)

For the definition of "subparts" see A279387.
For the triangle of subparts see A279391.
Row sums give A000203.
Row n has length A250068(n).
Cf. A001227, A005279, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239657, A244050, A245092, A250070, A261699, A262626.

A239665 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma of the smallest number whose symmetric representation of sigma has n parts.

Original entry on oeis.org

1, 2, 2, 5, 3, 5, 11, 5, 5, 11, 32, 12, 16, 12, 32, 74, 26, 14, 14, 26, 74, 179, 61, 29, 38, 29, 61, 179, 452, 152, 68, 32, 32, 68, 152, 452, 1250, 418, 182, 152, 100, 152, 182, 418, 1250, 3035, 1013, 437, 342, 85, 85, 342, 437, 1013, 3035, 6958, 1394, 638, 314, 154, 236, 154, 314, 638, 1394, 6958

Offset: 1

Omar E. Pol, Mar 23 2014

Row n is also row A239663(n) of A237270.

			----------------------------------------------------------------------
n    A239663(n)  Triangle begins:                        A266094(n)
----------------------------------------------------------------------
1        1       [1]                                         1
2        3       [2, 2]                                      4
3        9       [5, 3, 5]                                  13
4       21       [11, 5, 5, 11]                             32
5       63       [32, 12, 16, 12, 32]                      104
6      147       [74, 26, 14, 14, 26, 74]                  228
7      357       [179, 61, 29, 38, 29, 61, 179]            576
8      903       [452, 152, 68, 32, 32, 68, 152, 452]     1408
...
Illustration of initial terms:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.     _ _ 2          | |
.    |_ _|_ 2        | |
.     _ 1| |         | |
.    |_| |_|         |_|
.
For n = 2 we have that A239663(2) = 3 is the smallest number whose symmetric representation of sigma has 2 parts. Row 3 of A237593 is [2, 1, 1, 2] and row 2 of A237593 is [2, 2] therefore between both Dyck paths in the first quadrant there are two regions (or parts) of sizes [2, 2], so row 2 is [2, 2].
For n = 3 we have that A239663(3) = 9 is the smallest number whose symmetric representation of sigma has 3 parts. The 9th row of A237593 is [5, 2, 2, 2, 2, 5] and the 8th row of A237593 is [5, 2, 1, 1, 2, 5] therefore between both Dyck paths in the first quadrant there are three regions (or parts) of sizes [5, 3, 5], so row 3 is [5, 3, 5].

Cf. A000203, A005279, A196020, A236104, A237270, A237271, A235791, A237591, A237593, A239660, A239663, A239931-A239934, A240020, A240062, A244050, A245092, A262626, A266094.

a(16)-a(28) from Michel Marcus and Omar E. Pol, Mar 28 2014
a(29)-a(36) from Michel Marcus, Mar 28 2014
a(37)-a(45) from Michel Marcus, Mar 29 2014
a(46)-a(66) from Michel Marcus, Apr 02 2014

A253258 Square array read by antidiagonals, j>=1, k>=1: T(j,k) is the j-th number n such that the symmetric representation of sigma(n) has at least a part with maximum width k.

Original entry on oeis.org

1, 2, 6, 3, 12, 60, 4, 15, 72, 120, 5, 18, 84, 180, 360, 7, 20, 90, 240, 420, 840, 8, 24, 126, 252, 720, 1080, 3360, 9, 28, 140, 336, 1008, 1260, 3600, 2520, 10, 30, 144, 378, 1200, 1440, 3780, 5544, 5040, 11, 35, 168, 432, 1320, 1680, 3960, 6300, 7560, 10080, 13, 36, 198, 480, 1512, 1800, 4200, 6720, 9240, 12600, 15120

Offset: 1

Omar E. Pol, Jul 08 2015

This is a permutation of the natural numbers.
Row 1 gives A250070.
For more information about the widths of the symmetric representation of sigma see A249351 and A250068.
The next term: 120 < T(2,4) < 360.
From Hartmut F. W. Hoft, Sep 20 2024: (Start)
Column T(j,1), j>=1, forms A174905 and is a permutation of A357581. Numbers T(j,k), j>=1 and k>1, form A005279. Conjecture: Every column of the square array contains odd numbers.
The sequence of smallest odd numbers in each column forms A347980. E.g., in column 12 the smallest odd number is T(466, 12) = 765765 = A347980(12) which is equivalent to A250068(765765) = 12. (End)

			The corner of the square array T(j,k) begins:
  1,  6, 60, 120, 360, ...
  2, 12, 72, ...
  3, 15, 84, ...
  4, 18, ...
  5, 20, ...
  7, ...
  ...
For j = 1 and k = 2; T(1,2) is the first number n such that the symmetric representation of sigma(n) has a part with maximum width 2 as shown below:
.
      Dyck paths            Cells              Widths
      _ _ _ _             _ _ _ _
      _ _ _  |_          |_|_|_|_|_          / / / /
           |   |_              |_|_|_              / /
           |_ _  |             |_|_|_|             / / /
               | |                 |_|                 /
               | |                 |_|                 /
               | |                 |_|                 /
.
The widths of the symmetric representation of sigma(6) = 12 are [1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1], also the 6th row of triangle A249351.
From _Hartmut F. W. Hoft_, Sep 20 2024: (Start)
Extending the terms T(j,k) to a 12x12 square array:
j\k 1  2  3   4   5    6    7    8     9     10    11    12
--------------------------------------------------------------
1 | 1  6  60  120 360  840  3360 2520  5040  10080 15120 32760
2 | 2  12 72  180 420  1080 3600 5544  7560  12600 20160 36960
3 | 3  15 84  240 720  1260 3780 6300  9240  13860 25200 39600
4 | 4  18 90  252 1008 1440 3960 6720  10920 15840 35280 41580
5 | 5  20 126 336 1200 1680 4200 6930  11880 16380 40320 43680
6 | 7  24 140 378 1320 1800 4320 7140  14040 16800 42840 45360
7 | 8  28 144 432 1512 1980 4620 7920  16632 18480 46800 46200
8 | 9  30 168 480 1560 2016 4680 8190  17160 18900 47880 47520
9 | 10 35 198 504 1848 2100 5280 8400  17640 21420 56160 49140
10| 11 36 210 540 1890 2160 5400 9360  18720 21840 56700 51480
11| 13 40 216 594 2184 2340 5460 10296 19800 22680 57120 52920
12| 14 42 264 600 2310 2640 5940 10800 20790 23760 57960 54600
...
(End)

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..820
Index entries for sequences that are permutations of the natural numbers

Cf. A000203, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A249351, A250068, A250070, A250071.

Cf. A005279, A174905, A347980, A357581.

(* Computing table T(j,k) of size mxn with bound b *)
eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
mWidth[n_] :=Max[FoldList[#1+If[OddQ[#2], 1, -1]&, sDiv[n]]]
t253258[{m_, n_}, b_] := Module[{s=Table[0, {i, m+1}, {j, n}], k=1, w, f}, While[k<=b, w=mWidth[k]; If[w<=n, f=s[[m+1, w]]; If[fHartmut F. W. Hoft, Sep 20 2024 *)

More terms from Charlie Neder, Jan 11 2019

A053626 a(n) is the smallest positive integer k such that harmonic mean of n and k is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 10, 11, 4, 13, 14, 3, 16, 17, 6, 19, 5, 21, 22, 23, 8, 25, 26, 27, 4, 29, 6, 31, 32, 33, 34, 14, 12, 37, 38, 39, 10, 41, 7, 43, 44, 5, 46, 47, 16, 49, 50, 51, 52, 53, 18, 55, 8, 57, 58, 59, 12, 61, 62, 18, 64, 65, 6, 67, 68, 69, 28, 71, 9, 73, 74, 15, 76

Offset: 1

Henry Bottomley, Mar 20 2000

If a(n) <> n, then n is in A005279.

a(n) is the smallest positive integer k such that n + k divides n^2 + k^2. - Altug Alkan, Mar 29 2018

			a(6) = 2 because harmonic mean of 6 and 2 is 3 which is an integer and harmonic mean of 6 and 1 is 12/7 which is not an integer.

Altug Alkan, Table of n, a(n) for n = 1..10000
Wikipedia, Harmonic mean -- Two numbers

Cf. A005279.

Array[If[Count[Partition[Divisors[#], 2, 1], ?(#2 < 2 #1 & @@ # &)] == 0, #, Block[{k = 1}, While[! IntegerQ@ HarmonicMean@{k, #}, k++]; k]] &, 76] (* _Michael De Vlieger, Apr 05 2018 *)

a(n) = {my(k=1); while((2*n*k) % (n+k) != 0, k++); k; } \\ Altug Alkan, Mar 29 2018

Name edited by Altug Alkan, Mar 29 2018

A174903 Number of divisors d of n such that d

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 1, 6, 0, 0, 0, 3, 0, 3, 0, 0, 3, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 9, 0, 0, 1, 0, 0, 3, 0, 0, 0, 3, 0, 9, 0, 0, 2, 0, 1, 2, 0, 5, 0, 0, 0, 9, 0, 0, 0, 1, 0, 9, 1, 0, 0, 0, 0, 9, 0, 0, 1, 2, 0, 2, 0, 1, 4

Offset: 1

Reinhard Zumkeller, Apr 01 2010

nonn

			a(12) = #{(2,3), (3,4), (4,6)} = 3;
a(15) = #{(3,5)} = 1;
a(18) = #{(2,3), (6,9)} = 2;
a(20) = #{(4,5)} = 1;
a(24) = #{(2,3), (3,4), (4,6), (6,8), (8,12)} = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A005279, A174904, A174905.

import Data.List (intersect)
a174903 n = length [d | let ds = a027750_row n, d <- ds,
                    not $ null [e | e <- [d+1 .. 2*d-1] `intersect` ds]]
-- Reinhard Zumkeller, Sep 29 2014

a[n_] := Module[{d = Divisors[n]}, Count[d, ?(Length[Intersection[Range[# + 1, 2*# - 1], d]] > 0 &)]]; Array[a, 100] (* _Amiram Eldar, Apr 13 2024 *)

a(A174905(n)) = 0; a(A005279(n)) > 0.
a(A174904(n)) = n and a(m) <> n for m < A174904(n).
a(m)*a(n) <= a(m*n) for m, n coprime.

cp's OEIS Frontend

A020886 Ordered semiperimeters of primitive Pythagorean triangles.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A239657 Number of odd divisors m of n such that there is a divisor d of n with d < m < 2*d.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A174905 Numbers with no pair (d,e) of divisors such that d < e < 2*d.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

A279391 Irregular triangle read by rows in which row n lists the subparts of the successive layers of the symmetric representation of sigma(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A379288 Irregular triangle read by rows in which row n lists the odd divisors of n excluding odd divisors e for which there exists another divisor j with j < e < 2*j.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A279388 Irregular triangle read by rows: T(n,k) is the sum of the subparts in the k-th layer of the symmetric representation of sigma(n), if such a layer exists.

Views

Author

Keywords

Examples

Links

Crossrefs

A239665 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma of the smallest number whose symmetric representation of sigma has n parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A253258 Square array read by antidiagonals, j>=1, k>=1: T(j,k) is the j-th number n such that the symmetric representation of sigma(n) has at least a part with maximum width k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica