A071561 -id:A071561 - cp's OEIS frontend

A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.

1, 2, 3, 4, 5, 9, 6, 7, 15, 21, 8, 10, 25, 27, 63, 12, 11, 35, 33, 81, 147, 16, 13, 45, 39, 99, 171, 357, 18, 14, 49, 51, 117, 189, 399, 903, 20, 17, 50, 55, 153, 207, 441, 987, 2499, 24, 19, 70, 57, 165, 243, 483, 1029, 2709, 6069, 28, 22, 77, 65, 195, 261, 513, 1113

Offset: 1

Omar E. Pol, Apr 06 2014

This is a permutation of the positive integers.
All odd primes are in column 2 (together with some even composite numbers) because the symmetric representation of sigma(prime(i)) is [m, m], where m = (1 + prime(i))/2, for i >= 2.
The union of all odd-indexed columns gives A071562, the positive integers that have middle divisors. The union of all even-indexed columns gives A071561, the positive integers without middle divisors. - Omar E. Pol, Oct 01 2018
Each column in the table of A357581 is a subsequence of the respective column in the table of this sequence; however, the first row in the table of A357581 is not a subsequence of the first row in the table of this sequence. - Hartmut F. W. Hoft, Oct 04 2022
Conjecture: T(n,k) is the n-th positive integer with k 2-dense sublists of divisors. - Omar E. Pol, Aug 25 2025

			Array begins:
   1,  3,  9, 21,  63, 147, 357,  903, 2499, 6069, ...
   2,  5, 15, 27,  81, 171, 399,  987, 2709, 6321, ...
   4,  7, 25, 33,  99, 189, 441, 1029, 2793, 6325, ...
   6, 10, 35, 39, 117, 207, 483, 1113, 2961, 6783, ...
   8, 11, 45, 51, 153, 243, 513, 1197, 3025, 6875, ...
  12, 13, 49, 55, 165, 261, 567, 1239, 3087, 6909, ...
  16, 14, 50, 57, 195, 275, 609, 1265, 3249, 7011, ...
  18, 17, 70, 65, 231, 279, 621, 1281, 3339, 7203, ...
  20, 19, 77, 69, 255, 297, 651, 1375, 3381, 7353, ...
  24, 22, 91, 75, 273, 333, 729, 1407, 3591, 7581, ...
  ...
[Lower right hand triangle of array completed by _Hartmut F. W. Hoft_, Oct 04 2022]

Index entries for sequences that are permutations of the natural numbers

For programs see A237271 and A237593.
Row 1 is A239663.
Column 1 is A174973.
Columns 2..7: A239929, A279102, A280107, A320066, A320511, A357775.
For more information see A239663 and A239665.
Cf. A000203, A006254, A065091, A067742, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A239660, A239929, A239931-A239934, A245092, A262626, A319529, A319796, A319801, A319802.
Cf. A341969, A341970, A341971, A357581.

(* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
widthTable[n_, {r_, c_}] := Module[{k, list=Table[{}, c], parts}, For[k=1, k<=n, k++, parts=partsSRS[k]; If[parts<=c&&Length[list[[parts]]]=1, j--, vec[[PolygonalNumber[i+j-2]+j]]=arr[[i, j]]]]; vec]
a240062T[n_, r_] := TableForm[widthTable[n, {r, r}]]
a240062[6069, 10] (* data *)
a240062T[7581, 10] (* 10 X 10 array - Hartmut F. W. Hoft, Oct 04 2022 *)

a(n) > 128 from Michel Marcus, Apr 08 2014

A241008 Numbers n with the property that the number of parts in the symmetric representation of sigma(n) is even, and that all parts have width 1.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97

Offset: 1

Hartmut F. W. Hoft, Aug 07 2014

The first eight entries in A071561 but not in this sequence are 75, 78, 102, 105, 114, 138, 174 & 175.
The first eight entries in A239929 but not in this sequence are 21, 27, 33, 39, 51, 55, 57 & 65.
The union of this sequence and A241010 equals A174905 (see link in A174905 for a proof). Updated by Hartmut F. W. Hoft, Jul 02 2015
Let n = 2^m * Product_{i=1..k} p_i^e_i = 2^m * q with m >= 0, k >= 0, 2 < p_1 < ... < p_k primes and e_i >= 1, for all 1 <= i <= k. For each number n in this sequence k > 0, at least one e_i is odd, and for any two odd divisors f < g of n, 2^(m+1) * f < g. Let the odd divisors of n be 1 = d_1 < ... < d_2x = q where 2x = sigma_0(q). The z-th region of the symmetric spectrum of n has area a_z = 1/2 * (2^(m+1) - 1) *(d_z + d_(2x+1-z)), for 1 <= z <= 2x. Therefore, the sum of the area of the regions equals sigma(n). For a proof see Theorem 6 in the link of A071561. - Hartmut F. W. Hoft, Sep 09 2015, Sep 04 2018
First differs from A071561 at a(43). - Omar E. Pol, Oct 06 2018

Hartmut F. W. Hoft, Image for sigma(n) values

Cf. A000203, A071561, A071562, A174905, A236104, A237270, A237271, A237593, A238443, A241010, A246955.

(* path[n] and a237270[n] are defined in A237270 *)
atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]
Select[Range[100], atmostOneDiagonalsQ[#] && EvenQ[Length[a237270[#]]]&] (* data *)

A246955 Numbers j for which the symmetric representation of sigma(j) has two parts, each of width one.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 92, 94, 97, 101, 103, 106, 107, 109, 113, 116, 118, 122, 124, 127, 131, 134, 136, 137, 139, 142, 146, 148, 149, 151, 152, 157, 158, 163, 164, 166, 167, 172, 173, 178, 179, 181, 184, 188, 191, 193, 194, 197, 199

Offset: 1

Hartmut F. W. Hoft, Sep 08 2014

nonn

The sequence is the intersection of A239929 (sigma(j) has two parts) and of A241008 (sigma(j) has an even number of parts of width one).
The numbers in the sequence are precisely those defined by the formula for the triangle, see the link. The symmetric representation of sigma(j) has two parts, each part having width one, precisely when j = 2^(k - 1) * p where 2^k <= row(j) < p, p is prime and row(j) = floor((sqrt(8*j + 1) - 1)/2). Therefore, the sequence can be written naturally as a triangle as shown in the Example section.
The symmetric representation of sigma(j) = 2*j - 2 consists of two regions of width 1 that meet on the diagonal precisely when j = 2^(2^m - 1)*(2^(2^m) + 1) where 2^(2^m) + 1 is a Fermat prime (see A019434). This subsequence of numbers j is 3, 10, 136, 32896, 2147516416, ...[?]... (A191363).
The k-th column of the triangle starts in the row whose initial entry is the first prime larger than 2^(k+1) (that sequence of primes is A014210, except for 2).
Observation: at least the first 82 terms coincide with the numbers j with no middle divisors whose largest divisor <= sqrt(j) is a power of 2, or in other words, coincide with the intersection of A071561 and A365406. - Omar E. Pol, Oct 11 2023

			We show portions of the first eight columns, 0 <= k <= 7, of the triangle.
0    1    2     3     4     5     6     7
3
5    10
7    14
11   22   44
13   26   52
17   34   68    136
19   38   76    152
23   46   92    184
29   58   116   232
31   62   124   248
37   74   148   296   592
41   82   164   328   656
43   86   172   344   688
47   94   188   376   752
53   106  212   424   848
59   118  236   472   944
61   122  244   488   976
67   134  268   536   1072  2144
71   142  284   568   1136  2272
.    .    .     .     .     .
.    .    .     .     .     .
127  254  508   1016  2032  4064
131  262  524   1048  2096  4192  8384
137  274  548   1096  2192  4384  8768
.    .    .     .     .     .     .
.    .    .     .     .     .     .
251  502  1004  2008  4016  8032  16064
257  514  1028  2056  4112  8224  16448  32896
263  526  1052  2104  4208  8416  16832  33664
Since 2^(2^4) + 1 = 65537 is the 6543rd prime, column k = 15 starts with 2^15*(2^(2^16) + 1) = 2147516416 in row 6542 with 65537 in column k = 0.
For an image of the symmetric representations of sigma(m) for all values m <= 137 in the triangle see the link.
The first column is the sequence of odd primes, see A065091.
The second column is the sequence of twice the primes starting with 10, see A001747.
The third column is the sequence of four times the primes starting with 44, see A001749.
For related references also see A033676 (largest divisor of n less than or equal to sqrt(n)).

Hartmut F. W. Hoft, Equivalence proof of sequence and triangle
Hartmut F. W. Hoft, Image for sigma(n) values

Cf. A000203, A033676, A071561, A163280, A237270, A237271, A237593, A241008, A241010, A247687, A250068, A250070, A250071, A365406.

(* functions path[] and a237270[ ] are defined in A237270 *)
atmostOneDiagonalsQ[n_]:=SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], - 1] - path[n - 1], 1]]]
(* data *)
Select[Range[200], Length[a237270[#]]==2 && atmostOneDiagonalsQ[#]&]
(* function for computing triangle in the Example section through row 55 *)
TableForm[Table[2^k Prime[n], {n, 2, 56}, {k, 0, Floor[Log[2, Prime[n]] - 1]}], TableDepth->2]

Formula for the triangle of numbers associated with the sequence:

P(n, k) = 2^k * prime(n) where n >= 2, 0 <= k <= floor(log_2(prime(n)) - 1).

A368207 Bacher numbers: number of nonnegative representations of n = wx+yz with max(w,x) < min(y,z).

Original entry on oeis.org

1, 2, 2, 5, 3, 8, 4, 8, 9, 9, 6, 18, 7, 12, 14, 19, 9, 20, 10, 27, 16, 18, 12, 34, 20, 21, 20, 30, 15, 44, 16, 32, 24, 27, 30, 51, 19, 30, 28, 49, 21, 58, 22, 42, 41, 36, 24, 70, 35, 47, 36, 49, 27, 66, 36, 72, 40, 45, 30, 88, 31, 48, 62, 71, 42, 74, 34, 63, 48

Offset: 1

Don Knuth, Dec 16 2023

nonn

When n = p is an odd prime, Bacher proved that a(p) = (p+1)/2.
It appears that also a(k*p) = sigma(k)*(p+1)/2, for all prime p > 2k, where sigma(k) is the sum of the divisors of k (A000203).
It appears furthermore that a(p^2) = (p^2 + 3*p)/2; a(p^3) = (p^3 + p^2 + p + 1)/2; a(p^4) = (p^4 + p^3 + 3p^2 + p)/2, for all prime p.
Conjectures: (1) a(n) >= sigma(n)/2, with equality if and only if n has no middle divisors, i.e., if and only if n is in A071561. (2) a(n)/sigma(n) converges to 1/2. - Pontus von Brömssen, Dec 18 2023
From Chai Wah Wu, Dec 19 2023: (Start)
Considering representations where min(w,x)=0 shows that a(n) >= 2*A066839(n) - A038548(n).
It appears that a(p^5) = (p^5 + p^4 + p^3 + p^2 + p + 1)/2 for all prime p > 2 and a(p^6) = (p^6 + p^5 + p^4 + 3p^3 + p^2 + p)/2 for all prime p.
Conjecture: a(p^m) = sigma(p^m)/2 for odd m and all prime p > 2. a(p^m) = (sigma(p^m)-1)/2 + p^(m/2) for even m and all prime p. a(2^m) = sigma(2^m)/2 + 1/2 for m odd. (End)

			For n = 13, the a(13) = 7 solutions are (w,x,y,z) = (0,0,1,13), (0,0,13,1), (1,1,2,6), (1,1,3,4), (1,1,4,3), (1,1,6,2), (2,2,3,3).

Don Knuth, Table of n, a(n) for n = 1..10000
Roland Bacher, A quixotic proof of Fermat's two squares theorem for prime numbers, American Mathematical Monthly, Vol. 130, No. 9 (November 2023), 824-836; arXiv version, arXiv:2210.07657 [math.NT], 2022.
Michael S. Branicky, Python translation of Knuth's CWEB program
Pontus von Brömssen, Plot of (n, a(n) - sigma(n)/2) for n = 1..100000.
Pontus von Brömssen, Plot of (n, b(n)) for n = 1..100000, where a(n) = sigma(n)/2 + b(n)*sqrt(n).
Don Knuth, CWEB program.
Peter Luschny, A Formula for the Bacher Numbers.

Cf. A000203, A071561, A368276 (monotone), A368341 (fixed points), A368457, A368458 (semiprimes), A368580 (degenerated).

CWEB
```
@ See Knuth link.
```

function A368207(n)
    t(n) = (d for d in divisors(n) if d * d <= n)
    s(d) = d * d == n ? d * 2 - 1 : d * 4 - 2
    c(y, w, wx) = max(1, 2*(Int(w*w < wx) + Int(y*y < n - wx)))
    sum(sum(sum(c(y, w, wx) for y in t(n - wx) if wx < y * w; init=0)
    for w in t(wx)) for wx in 1:div(n, 2); init=sum(s(d) for d in t(n)))
end
println([A368207(n) for n in 1:69])  # Peter Luschny, Dec 21 2023

t[n_] := t[n] = Select[Divisors[n], #^2 <= n&];
A368207[n_] := Sum[(1 + Boole[d^2 < n])(2d - 1),{d, t[n]}] + Sum[If[wx < y*w, Max[1, 2(Boole[w^2 < wx] + Boole[y^2 < n-wx])], 0], {wx, Floor[n/2]},{w, t[wx]}, {y, t[n - wx]}];
Array[A368207, 100] (* Paolo Xausa, Jan 02 2024 *)

# See Branicky link for translation of Knuth's CWEB program.

from math import isqrt
def A368207(n):
    c, r = 0, isqrt(n)
    for w in range(r+1):
        for x in range(w,r+1):
            wx = w*x
            if wx>n:
                break
            for y in range(x+1,r+1):
                for z in range(y,n+1):
                    yz = wx+y*z
                    if yz>n:
                        break
                    if yz==n:
                        m = 1
                        if w!=x:
                            m<<=1
                        if y!=z:
                            m<<=1
                        c+=m
    return c # Chai Wah Wu, Dec 19 2023

from sympy import divisors
# faster program
def A368207(n):
    c = 0
    for d2 in divisors(n):
        if d2**2 > n:
            break
        c += (d2<<2)-2 if d2**2>1)+1):
        for d1 in divisors(wx):
            if d1**2 > wx:
                break
            for d2 in divisors(m:=n-wx):
                if d2**2 > m:
                    break
                if wx < d1*d2:
                    k = 1
                    if d1**2 != wx:
                        k <<=1
                    if d2**2 != m:
                        k <<=1
                    c+=k
    return c # Chai Wah Wu, Dec 19 2023

Let t(n) = {d|n and d*d <= n}, and s(d, n) = 2*d - 1 if d*d = n, otherwise 4*d - 2. Then a(n) = (Sum_{d in t(n)} s(d, n)) + (Sum_{k=1..floor(n/2)} Sum_{w in t(k)} Sum_{y in t(n-k) and k < y*w} max(1, 2*([w*w < k] + [y*y < n - k]))), where [] denote the Iverson brackets. (See the 'Julia' implementation below.) - Peter Luschny, Dec 21 2023

A241558 Smallest part of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 3, 2, 7, 3, 12, 4, 15, 3, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 5, 18, 12, 60, 5, 21, 6, 56, 15, 72, 16, 63, 7, 27, 12, 91, 19, 30, 8, 90, 21, 96, 22, 42, 23, 36, 24, 124, 7, 15, 10, 49, 27, 120, 8, 120, 11, 45, 30, 168, 31, 48, 12, 127, 9, 144, 34, 63, 13

Offset: 1

Michel Marcus and Omar E. Pol, Apr 29 2014

nonn

If A237271(n) = 1 then a(n) = A241559(n) = A241838(n) = A000203(n).
If n is an odd prime then a(n) = (n + 1)/2 = A241559(n) = A241838(n).
For more information see A237270 and A237593.

			For n = 9 the symmetric representation of sigma(9) = 13 in the first quadrant looks like this:
y
.
._ _ _ _ _ 5
|_ _ _ _ _|
.         |_ _ 3
.         |_  |
.           |_|_ _ 5
.               | |
.               | |
.               | |
.               | |
. . . . . . . . |_| . . x
.
There are three parts [5, 3, 5] and the smallest part is 3 so a(9) = 3.
For n = 45 the symmetric representation of sigma(45) = 78 has three parts [23, 32, 23] and the smallest part is 23 so a(45) = 23.
For n = 63 the symmetric representation of sigma(63) = 104 has five parts [32, 12, 16, 12, 32] and the smallest part is 12 so a(63) = 12.

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A000203, A071561, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A241559, A241838, A245092, A262626.

(* Function a237270[] is defined in A237270 *)
a241558[n_]:=Min[a237270[n]]
Map[a241558,Range[64]] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)

More terms from Jinyuan Wang, Feb 14 2020

A262259 Numbers k such that the symmetric representation of sigma(k) has only two parts and they meet at the center of the Dyck path.

Original entry on oeis.org

3, 10, 78, 136, 666, 820, 1830, 2628, 4656, 5886, 6328, 16290, 18528, 28920, 32896, 39340, 48828, 56616, 62128, 78606, 80200, 83436, 88410, 93528, 100576, 104196, 135460, 146070, 166176, 180300, 187578, 190036

Offset: 1

Hartmut F. W. Hoft, Sep 16 2015

For a proof of the formula see the link and also the links in A239929 and A071561. This formula allows for a fast computation of numbers in the sequence that does not require computations of Dyck paths.
Subsequence of A239929.
A191363 is a subsequence.
All terms are triangular numbers.
More precisely, all terms are second hexagonal numbers (A014105). There are no terms with middle divisors. - Omar E. Pol, Oct 31 2018
Numbers k such that the concatenation of the widths of the symmetric representation of sigma(k) is a cyclops numbers (A134808). - Omar E. Pol, Aug 29 2021

			q = 128 = 2^7 is the 15th term in A174973 for which 2*n+1 = 2^8 + 1 is prime so that a(15) = 2^7 * (2^8 + 1) = 32896. The two parts in the symmetric representation of sigma of a(15) have width 1 and sigma(a(15)) = 2 * a(15) - 2.
q = 308 is the 32nd term in A174973 for which 2*n+1 is prime so that a(32) = 308 * 617 = 190036. The maximum width of the two regions is 2 and sigma(a(32)) = 415296.
For n = 21, the symmetric representation of sigma(21) has two parts that meet at the center of the Dyck path, but 21 is not in the sequence because the symmetric representation of sigma(21) has more than two parts. - _Omar E. Pol_, Sep 18 2015
From _Omar E. Pol_, Oct 05 2015: (Start)
Illustration of initial terms (n = 1, 2):
. y
.  |
.  |_ _ _ _ _ _
.  |_ _ _ _ _  |
.  |         | |_
.  |         |_ _|_
.  |             | |_ _
.  |             |_ _  |
.  |                 | |
.  |_ _              | |
.  |_ _|_            | |
.  |   | |           | |
.  |_ _|_|_ _ _ _ _ _|_|_ _ x
.       3             10
.
The symmetric representation of sigma(3) = 2 + 2 = 4 has two parts and they meet at the point (2, 2), so a(1) = 3.
The symmetric representation of sigma(10) = 9 + 9 = 18 has two parts and they meet at the point (7, 7), so a(2) = 10.
(End)
Also 10 is in the sequence because the concatenation of the widths of the symmetric representation of sigma(10) is 1111111110111111111 and it is a cyclops number (A134808). - _Omar E. Pol_, Aug 29 2021

Hartmut F. W. Hoft, proof of formula

Cf. A000203, A000217, A014105, A071561, A134808, A174973, A191363, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239929, A245092, A249351 (widths), A262045, A262048, A262626.

(* test for membership in A174973 *)
a174973Q[n_]:=Module[{d=Divisors[n]}, Select[Rest[d] - 2 Most[d], #>0&]=={}]
a174973[n_]:=Select[Range[n], a174973Q]
(* compute numbers in the sequence *)
a262259[n_]:=Map[#(2#+1)&, Select[a174973[1, n], PrimeQ[2#+1]&]]
a262259[308] (* data *)

Terms are equal to q*(2*q + 1) where q is in A174973 and 2*q + 1 is prime.

A319802 Even numbers without middle divisors.

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, 230, 232, 236, 244, 246, 248, 250, 254, 258, 262, 268, 274, 278, 282, 284

Offset: 1

Omar E. Pol, Sep 28 2018

nonn

Even numbers k such that the symmetric representation of sigma(k) has an even number of parts.
For the definition of middle divisors, see A067742.
For more information about the symmetric representation of sigma(k) see A237593.
Let p be a prime > 5. Then a(n) is a number of the form m*p where m is an even number < sqrt(p). - David A. Corneth, Sep 28 2018
First differs from A244894 at a(51) = 230. - R. J. Mathar, Oct 04 2018
Is this twice A101550? - Omar E. Pol, Oct 04 2018
This sequence is not twice A101550: first differs at a(57) = 250 != 254 = 2*A101550(57). - Michael S. Branicky, Oct 14 2021

			10 is in the sequence because it's an even number and the symmetric representation of sigma(10) = 18 has an even number of parts as shown below:
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Intersection of A005843 and A071561.
Cf. A244894 (a subsequence).
Cf. A000203, A067742, A071090, A071562, A236104, A237270, A237271, A237593, A239932, A239934, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319796, A319801.

from sympy import divisors
def ok(n):
    if n < 2 or n%2 == 1: return False
    return not any(n//2 <= d*d < 2*n for d in divisors(n, generator=True))
print(list(filter(ok, range(285)))) # Michael S. Branicky, Oct 14 2021

A340833 a(n) is the number of vertices in the diagram of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 6, 7, 10, 9, 12, 11, 14, 14, 15, 13, 18, 13, 17, 20, 22, 15, 22, 15, 22, 23, 21, 17, 26, 22, 21, 25, 28, 19, 30, 19, 30, 27, 23, 26, 32, 21, 25, 29, 34, 21, 34, 21, 33, 36, 27, 23, 38, 30, 38, 31, 35, 23, 38, 35, 42, 33, 29, 25, 42, 25, 29, 42, 42, 37, 44, 27

Offset: 1

Omar E. Pol, Jan 23 2021

If A237271(n) is odd then a(n) is even.
If A237271(n) is even then a(n) is odd.
The above sentences arise that the diagram is always symmetric for any value of n hence the number of edges is always an even number. Also from Euler's formula.
Indices of odd terms give A071561.
Indices of even terms give A071562.
For another version with subparts see A340847 from which first differs at a(6).
The parity of this sequence is also the characteristic function of numbers that have no middle divisors (cf. A348327). - Omar E. Pol, Oct 14 2021

			Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             |   |_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      6        7         10           9              12
.
For n = 6 the diagram has 12 vertices so a(6) = 12.
On the other hand the diagram has 12 edges and only one part or region, so applying Euler's formula we have that a(6) = 12 - 1 + 1 = 12.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
n:              7                    8                      9
a(n):          11                   14                     14
.
For n = 9 the diagram has 14 vertices so a(9) = 14.
On the other hand the diagram has 16 edges and three parts or regions, so applying Euler's formula we have that a(9) = 16 - 3 + 1 = 14.
Another way for the illustration of initial terms is as follows:
--------------------------------------------------------------------------
.  n  a(n)                             Diagram
--------------------------------------------------------------------------
            _
   1   4   |_|  _
              _| |  _
   2   6     |_ _| | |  _
                _ _|_| | |  _
   3   7       |_ _|  _| | | |  _
                  _ _|  _| | | | |  _
   4  10         |_ _ _|  _|_| | | | |  _
                    _ _ _|  _ _| | | | | |  _
   5   9           |_ _ _| |    _| | | | | | |  _
                      _ _ _|  _|  _|_| | | | | | |  _
   6  12             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  11               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  14                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  14                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|    _ _| | | | | | | | |
  10  15                     |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |      _| |  _ _ _| | | | | |
  11  13                       |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  18                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |    _| |  _ _ _|
  13  13                           |_ _ _ _ _ _ _| | |  _|  _|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  17                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  20                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  22                                 |_ _ _ _ _ _ _ _ _|
...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n) for n=1..10^4, accentuating a(m) for m=1..2^8 for clarity, and labeling a(k) for k=1..24.

Parity gives A348327.
Cf. A237271 (number of parts or regions).
Cf. A340846 (number of edges).
Cf. A340847 (number of vertices in the diagram with subparts).
Cf. A294723 (total number of vertices in the unified diagram).
Cf. A239931-A239934 (illustration of first 32 diagrams).
Cf. A000203, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A237590, A237591, A237593, A239660, A245092, A262626, A340848.

MapAt[# + 1 &, #, 1] &@ Map[Length@ Union[Join @@ #] - 1 &, Partition[Prepend[#, {{0, 0}}], 2, 1]] &@ Table[{{0, 0}}~Join~Accumulate[Join[#, Reverse[Reverse /@ (-1*#)]]] &@ MapIndexed[Which[#2 == 1, {#1, 0}, Mod[#2, 2] == 0, {0, #1}, True, {-#1, 0}] & @@ {#1, First[#2]} &, If[Length[#] == 0, {n, n}, Join[{n}, #, {n - Total[#]}]]] &@ Differences[n - Array[(Ceiling[(n + 1)/# - (# + 1)/2]) &, Floor[(Sqrt[8 n + 1] - 1)/2]]], {n, 67}] (* Michael De Vlieger, Oct 27 2021 *)

a(n) = A340846(n) - A237271(n) + 1 (Euler's formula).

Terms a(33) and beyond from Michael De Vlieger, Oct 27 2021

A241838 Column 1 of A237270, also the right border.

Original entry on oeis.org

1, 3, 2, 7, 3, 12, 4, 15, 5, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 11, 18, 12, 60, 13, 21, 14, 56, 15, 72, 16, 63, 17, 27, 18, 91, 19, 30, 20, 90, 21, 96, 22, 42, 23, 36, 24, 124, 25, 39, 26, 49, 27, 120, 28, 120, 29, 45, 30, 168, 31, 48, 32, 127

Offset: 1

Omar E. Pol, Apr 29 2014

First differs from A241559 at a(45).
If A237271(n) = 1 then a(n) = A241558(n) = A241559(n) = A000203(n).
If n is an odd prime then a(n) = (n + 1)/2 = A241558(n) = A241559(n).
For more information see A237593.

			For n = 45 the symmetric representation of sigma(45) = 78 has three parts [23, 32, 23], both the first and the last term are equal to 23, so a(45) = 23.

Robert Price, Table of n, a(n) for n = 1..10000

Cf. A000203, A071561, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A241558, A241559, A245092, A262626.

Map[First[a237270[#]]&,Range[64]] (* data : computing all parts *)
(* computing only the first part of the symmetric representation of sigma(n) *)
row[n_] := Floor[(Sqrt[8n+1]-1)/2] (* in A237591 *)
f[n_, k_] := If[Mod[n-k*(k+1)/2, k]==0, (-1)^(k+1), 0]
g[n_, k_] := Ceiling[(n+1)/k-(k+1)/2] - Ceiling[(n+1)/(k+1)-(k+2)/2] (* in A237591 *)
a241838[n_] := Module[{r=row[n], widths={}, i=1, w=0, len, legs}, w+=f[n, i]; While[i<=r && w!=0, AppendTo[widths, w]; i++; w+=f[n, i]]; len=Length[widths]; legs=Map[g[n, #]&, Range[len]]; If[lenHartmut F. W. Hoft, Jan 25 2018 *)

a(n) = A237270(n, 1) = A237270(n, A237271(n)).

A241559 Largest part of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 3, 2, 7, 3, 12, 4, 15, 5, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 11, 18, 12, 60, 13, 21, 14, 56, 15, 72, 16, 63, 17, 27, 18, 91, 19, 30, 20, 90, 21, 96, 22, 42, 32, 36, 24, 124, 25, 39, 26, 49, 27, 120, 28, 120, 29, 45, 30, 168, 31, 48, 32, 127

Offset: 1

Michel Marcus and Omar E. Pol, Apr 29 2014

nonn

First differs from A241838 at a(45).
If A237271(n) = 1 then a(n) = A241558(n) = A241838(n) = A000203(n).
If n is an odd prime then a(n) = (n + 1)/2 = A241558(n) = A241838(n).
For more information see A237593.

			For n = 9 the symmetric representation of sigma(9) = 13 in the first quadrant looks like this:
y
.
._ _ _ _ _ 5
|_ _ _ _ _|
.         |_ _ 3
.         |_  |
.           |_|_ _ 5
.               | |
.               | |
.               | |
.               | |
. . . . . . . . |_| . . x
.
There are three parts [5, 3, 5] and the largest part is 5 so a(9) = 5.
For n = 45 the symmetric representation of sigma(45) = 78 has three parts [23, 32, 23] and the largest part is 32 so a(45) = 32.

Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A241558, A071561, A071562, A241838.

(* Function a237270[] is defined in A237270 *)
a241559[n_]:=Max[a237270[n]]
Map[a241559,Range[64]] (* data *)
(* Hartmut F. W. Hoft, Sep 19 2014 *)

cp's OEIS Frontend

A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.

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A241008 Numbers n with the property that the number of parts in the symmetric representation of sigma(n) is even, and that all parts have width 1.

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A246955 Numbers j for which the symmetric representation of sigma(j) has two parts, each of width one.

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A368207 Bacher numbers: number of nonnegative representations of n = w*x+y*z with max(w,x) < min(y,z).

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A241558 Smallest part of the symmetric representation of sigma(n).

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A262259 Numbers k such that the symmetric representation of sigma(k) has only two parts and they meet at the center of the Dyck path.

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A319802 Even numbers without middle divisors.

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A368207 Bacher numbers: number of nonnegative representations of n = wx+yz with max(w,x) < min(y,z).