A298856 -id:A298856 - cp's OEIS frontend

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

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

Offset: 1

Hartmut F. W. Hoft, Sep 04 2018

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

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

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

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

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

A279029 Numbers k with the property that the smallest and the largest Dyck path of the symmetric representation of sigma(k) do not share line segments.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 136, 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

Omar E. Pol, Dec 08 2016

nonn

Numbers k such that the symmetric representation of sigma(k) is formed by only one part, or that it's formed by only two parts and they meet at the center.
Numbers k whose total length of all line segments of the symmetric representation of sigma(k) is equal to 4*k (cf. A348705). For the positive integers k that are not in this sequence the mentioned total length is < 4*k. - Omar E. Pol, Nov 02 2021
From Hartmut F. W. Hoft, Jan 25 2025: (Start)
The following three statements are equivalent for numbers k >= 1:
(1) The symmetric representation of sigma(k) is formed of 2 parts that meet at the diagonal.
(2) A249223(k, A003056(k)) = 0 is the only 0 in row k of the triangle, and A237591(k, A003056(k)) = 1.
(3) Row k of the triangle in A341969 contains a single 0 at the center position.
The following two statements are equivalent for numbers k >= 1:
(1) The symmetric representation of sigma(k) consists of a single part.
(2) Row k of the triangle in A249223 contains no 0. (End)
This sequence is the disjoint union of A262259 and A174973. Each member of A262259 has the form k = q*(2*q + 1) where 2*q + 1 is prime; also A003056(k) = 2*q. Therefore [q, 2*q] contains a divisor q of k while (q, 2*q] contains no divisor of k. A262259 is a subsequence of A298259, see also A240542. - Hartmut F. W. Hoft, Mar 24 2025
My two links below give detailed proofs for the last comment.  - Hartmut F. W. Hoft, Jun 10 2025

			1, 2, 3, 4, 6, 8, 10, 12 and 16 are in the sequence because the smallest and the largest Dyck path of their symmetric representation of sigma do not share line segments, as shown below.
Illustration of initial terms:
  n
  .      _ _ _ _   _   _   _   _       _
  1     |_| | | | | | | | | | | |     | |
  2     |_ _|_| | | | | | | | | |     | |
  3     |_ _|  _|_| | | | | | | |     | |
  4     |_ _ _|    _|_| | | | | |     | |
         _ _ _|  _|  _ _|_| | | |     | |
  6     |_ _ _ _|  _| |  _ _|_| |     | |
         _ _ _ _| |_ _|_|    _ _|     | |
  8     |_ _ _ _ _|  _|     |    _ _ _| |
         _ _ _ _ _| |      _|   |  _ _ _|
  10    |_ _ _ _ _ _|  _ _|    _| |
         _ _ _ _ _ _| |      _|  _|
  12    |_ _ _ _ _ _ _|  _ _|  _|
                        |  _ _|
                        | |
         _ _ _ _ _ _ _ _| |
  16    |_ _ _ _ _ _ _ _ _|
  ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Hartmut F. W. Hoft, Proof: Dyck Paths of SRS(n) meeting only at diagonal
Hartmut F. W. Hoft, Proofs for A279029 = A174973 union A262259

UNION of A174973 and A262259.
Positions of 0's in A279228.
Complement is A279244.
Cf. A000203, A003056, A008586, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A249223, A262259, A323648, A348705, A379968 (characteristic function).
Cf. A240542, A298856.

(* Function path[] is defined in A237270 *)
a279029Q[n_] := Length[Select[Transpose[{Take[path[n], {2,-2}], path[n-1]}], #[[1]]==#[[2]]&]]<=1
a279029[n_] := Select[Range[n], a279029Q]
a279029[256]
(* Alternate, faster function based on A249223 *)
a003056[n_] := Floor[(Sqrt[8n+1]-1)/2]
t249223[n_] :=FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, a003056[n]]]
a262259Q[n_] := Position[t249223[n], 0]=={{a003056[n]}}&&Last[t237591[n]]==1
a174973Q[n_] := !MemberQ[t249223[n], 0]
a279029[n_] := Select[Range[n], a262259Q[#]||a174973Q[#]&]
a279029[256] (* Hartmut F. W. Hoft, Jan 25 2025 *)

is_A279029 = A379968; \\ Antti Karttunen, Jan 12 2025

a(n) = A323648(n-1) + 1, for n >= 2. - Hartmut F. W. Hoft, Jan 25 2025

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

Original entry on oeis.org

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

Offset: 1

Hartmut F. W. Hoft, Jan 27 2018

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

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

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

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

A368945 a(n) is the length of the central extent of width 0 for the symmetric representation of sigma, SRS(n), when that has an even number of parts otherwise -1.

Original entry on oeis.org

-1, -1, 0, -1, 2, -1, 4, -1, -1, 0, 8, -1, 10, 2, -1, -1, 14, -1, 16, -1, 0, 6, 20, -1, -1, 8, 2, -1, 26, -1, 28, -1, 4, 12, -1, -1, 34, 14, 6, -1, 38, -1, 40, 2, -1, 18, 44, -1, -1, -1, 10, 4, 50, -1, 0, -1, 12, 24, 56, -1, 58, 26, -1, -1, 2, -1, 64, 8, 16, -1, 68, -1, 70, 32, 4, 10, -1, 0, 76, -1

Offset: 1

Hartmut F. W. Hoft, Jan 10 2024

sign

The central extent of width 0 for SRS(n) is that uninterrupted section where both boundary Dyck paths coincide, includes the point on the diagonal and thus has even length.
The maximum possible extent of width 0 in SRS(n) for odd numbers n is 2n - (n+1) - 2 = n - 3. This is achieved only by odd prime numbers p so that the values a(p) = p - 3 form the subsequence of records in this sequence; in particular, p-3 is the largest instantiated width 0 extent in the interval 1..p.
Conjecture: Every nonnegative even number occurs in this sequence.

			a(3) = 3 - 3 = 0 and a(2*5) = 5 - 5 = 0 by the formulas (see the illustrations in A237270 and A237593).
a(27) = 2 since SRS(a(27)) consists of the 4 parts of width 1 that have lengths 14, 6, 6, 14 and leave a section of length 2 across the diagonal of width 0 (see the illustration in A237593).

Cf. A071561, A071562, A235791, A237048, A237270, A237593, A249223, A298856.

t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]] (* row n in triangle of A249223 *)
extent0[n_] := Module[{t=t249223[n], s}, s=Position[t, 1][[-1, -1]]; If[Last[t]==0, 2Ceiling[(n+1)/(s+1)-(s+1)/2]-2, -1]]a368945[n_] := Map[extent0, Range[n]]a368945[80]

a(n) = 2*ceiling((n+1)/(s(n)+1) - (s(n)+1)/2) - 2, where s(n) = position of the rightmost 1 in row n of the triangle in sequence A249223 when the last entry in that row is 0, and a(n) = -1 otherwise.
a(p) = p-3, p >= 3 prime, since s(p) = 1.
a(2*p) = p-5, p >= 5 prime, since s(p) = 3.
a(A071561(n)) >= 0, a(A071562(n)) = -1, and a(A298856(n)) = 0 for all n >= 1.

A361209 Second hexagonal numbers having middle divisors.

Original entry on oeis.org

36, 210, 300, 528, 990, 1176, 1485, 1596, 2080, 2346, 3240, 3570, 4095, 4278, 4851, 5460, 6555, 6786, 7260, 8256, 8778, 9870, 10440, 11628, 12880, 13530, 14196, 14535, 15225, 15576, 17020, 17766, 20100, 20910, 21736, 22578, 23436, 24310, 25200, 26565, 27495, 27966, 30876

Offset: 1

Omar E. Pol, Mar 10 2023

nonn

The middle divisors of n are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).

Also numbers k with the property that in the symmetric representation of sigma(k) the smallest Dyck path has a central peak and the largest Dyck path has a central valley and both Dyck paths do not meet in the center.

			36 is in the sequence because it is a second hexagonal number (A014105) and it has a middle divisor, the 6.
On the other hand the 35th row of A237593 is [18,7,3,2,2,1,2,2,1,2,2,3,7,18] and the 36th row of the same triangle is [19,6,4,2,2,1,1,1,1,1,1,2,2,4,6,19]. Since the smallest Dyck path of the symmetric representation of sigma(36) has a central peak and the largest Dyck path has a central valley and both Dyck paths do not meet in the center so 36 is in the sequence. The diagram is too large to include.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Intersection of A014105 and A071562.
Nonzero terms of A014107 without the terms of A298856.
Cf. A067742, A236104, A237048, A237591, A237593, A262626, A240542.

A071562Q[n_]:=With[{m1=Sqrt[n/2],m2=Sqrt[2n]},DivisorSum[n,#&,m1<=#0];
With[{upto=200},Select[Array[#(2#+1)&,upto],A071562Q]] (* Checks the first 200 second hexagonal numbers *) (* Paolo Xausa, Oct 23 2023 *)

hasmd(n)=fordiv(n, d, if(d^2>=n/2 && d^2<2*n, return(1))); 0; \\ A014105
select(hasmd, vector(150, n, n*(2*n + 1))) \\ Michel Marcus, Mar 10 2023

More terms from Michel Marcus, Mar 10 2023

cp's OEIS Frontend

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

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A279029 Numbers k with the property that the smallest and the largest Dyck path of the symmetric representation of sigma(k) do not share line segments.

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

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A368945 a(n) is the length of the central extent of width 0 for the symmetric representation of sigma, SRS(n), when that has an even number of parts otherwise -1.

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A361209 Second hexagonal numbers having middle divisors.

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