A060831 -id:A060831 - cp's OEIS frontend

A285901 Total number of partitions of all positive integers <= n into an odd number of consecutive parts.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 20, 21, 22, 24, 25, 27, 29, 30, 31, 33, 35, 36, 38, 40, 41, 44, 45, 46, 48, 49, 52, 54, 55, 56, 58, 60, 61, 64, 65, 66, 70, 71, 72, 74, 76, 78, 80, 81, 82, 85, 87, 89, 91, 92, 93, 96, 97, 98, 102, 103, 105, 108, 109, 110, 112, 115, 116, 119, 120, 121, 124, 125, 128, 130

Offset: 1

Omar E. Pol, May 02 2017

nonn

a(n) is also the total number of odd divisors of k less than sqrt(2*k), for k = 1..n.
Conjecture: a(n) is also the total number of subparts present (totally or partially) in an octant of the symmetric representations of sigma of all positive integers <= n.
For more information about the "subparts" of the symmetric representation of sigma see A279387 and A237593.

Partial sums of A082647.

Cf. A001227, A060831, A131576, A196020, A235791, A236104, A237048, A237591, A237593, A244250, A262618, A279387, A285902.

a(n) = A060831(n) - A285902(n).

A285900 Sum of all parts of all partitions of all positive integers <= n into consecutive parts.

Original entry on oeis.org

1, 3, 9, 13, 23, 35, 49, 57, 84, 104, 126, 150, 176, 204, 264, 280, 314, 368, 406, 446, 530, 574, 620, 668, 743, 795, 903, 959, 1017, 1137, 1199, 1231, 1363, 1431, 1571, 1679, 1753, 1829, 1985, 2065, 2147, 2315, 2401, 2489, 2759, 2851, 2945, 3041, 3188, 3338, 3542, 3646, 3752, 3968, 4188, 4300, 4528, 4644, 4762, 5002

Offset: 1

Omar E. Pol, May 02 2017

nonn

a(n) is also the sum of all parts of all partitions of all positive integers <= n into an odd number of equal parts. - Omar E. Pol, Jun 05 2017

			For n = 15, there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. The sum of all parts of these four partitions is 60, and a(14) = 204, so a(15) = 204 + 60 = 264.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Partial sums of A245579.

Cf. A001227, A060831, A204217, A237593, A285891, A285899.

a285900[n_] := Accumulate[Map[# DivisorSum[#, 1 &, OddQ] &, Range[n]]]
a285900[60] (* data *) (* Hartmut F. W. Hoft, Jun 06 2017 *)

a(n)=sum(i=1, n, i * sumdiv(i, d, d%2)); \\ Andrew Howroyd, Nov 06 2018

a(n)=sum(k=1, (n+1)\2, (2*k - 1)/2 * (n\(2*k - 1)) * (1 + n\(2*k - 1))); \\ Andrew Howroyd, Nov 06 2018

a(n) = Sum_{k=1..floor((n+1)/2)} (2*k-1)/2 * floor(n/(2*k-1)) * floor(1 + n/(2*k-1)). - Daniel Suteu, Nov 06 2018

A263086 Partial sums of A099777, where A099777(n) gives the number of divisors of n-th even number.

Original entry on oeis.org

2, 5, 9, 13, 17, 23, 27, 32, 38, 44, 48, 56, 60, 66, 74, 80, 84, 93, 97, 105, 113, 119, 123, 133, 139, 145, 153, 161, 165, 177, 181, 188, 196, 202, 210, 222, 226, 232, 240, 250, 254, 266, 270, 278, 290, 296, 300, 312, 318, 327, 335, 343, 347, 359, 367, 377, 385, 391, 395, 411, 415, 421, 433, 441, 449, 461, 465, 473, 481

Offset: 1

Antti Karttunen, Oct 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10082
A. Karttunen, Ratio a(n)/A263085(n) drawn with OEIS Plot2-script

Cf. A000005, A006218, A099777, A263084, A263085, A060831.

Cf. also A262518, A262519.

with(numtheory): seq(add(tau(2*k), k=1..n), n= 1..60); # Ridouane Oudra, Aug 24 2019

Accumulate[DivisorSigma[0, 2 Range@ 69]] (* Michael De Vlieger, Oct 13 2015 *)

a(n) = sum(k=1, n, numdiv(2*k)); \\ Michel Marcus, Aug 25 2019

from math import isqrt
def A263086(n): return (t:=isqrt(m:=n>>1))**2-((s:=isqrt(n))**2<<1)+((sum(n//k for k in range(1,s+1))<<1)-sum(m//k for k in range(1,t+1))<<1) # Chai Wah Wu, Oct 23 2023

a(1) = 2; for n > 1, a(n) = A000005(2*n) + a(n-1) [where A000005(k) gives the number of divisors of k].
Other identities. For all n >= 1:
a(n) = A263084(n) + A263085(n).
a(n) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019
From Ridouane Oudra, Aug 24 2019: (Start)
a(n) = Sum_{k=1..n} A000005(2*k)
a(n) = A006218(n) + A060831(n). (End)

A069930 Number of integers of the form (n+k)/(n-k) with 1 <= k <= n-1.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 2, 3, 4, 4, 2, 6, 2, 4, 6, 4, 2, 7, 2, 6, 6, 4, 2, 8, 4, 4, 6, 6, 2, 10, 2, 5, 6, 4, 6, 10, 2, 4, 6, 8, 2, 10, 2, 6, 10, 4, 2, 10, 4, 7, 6, 6, 2, 10, 6, 8, 6, 4, 2, 14, 2, 4, 10, 6, 6, 10, 2, 6, 6, 10, 2, 13, 2, 4, 10, 6, 6, 10, 2, 10, 8, 4, 2, 14, 6, 4, 6, 8, 2, 16, 6, 6, 6, 4, 6

Offset: 1

Benoit Cloitre, May 05 2002

Number of r X s integer-sided rectangles such that r < s, r + s = 2n and r | s. - Wesley Ivan Hurt, Apr 24 2020

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A032741, A069283, A000005 (tau), A001227, A006218, A060831, A066743.

A066660(n) - 1.

Table[Count[Table[(n+k)/(n-k),{k,n-1}],?IntegerQ],{n,100}] (* _Harvey P. Dale, Jun 04 2019 *)

for(n=1,150,print1(sum(i=1,n-1,if((n+i)%(n-i),0,1)),","))

{a(n) = if( n<1, 0, numdiv(2*n) - 2)} /* Michael Somos, Aug 30 2012 */

a(n) = A032741(n) + A069283(n) = A000005(n) - 1 + A001227(n) - 1 = tau(n) + A001227(n) - 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 13 2002
Asymptotic formula: since sum(k=1, n, a(k)) = sum(k=1, n, tau(k)) + sum(k=1, n, A001227(k)) - 2*n = A006218(n) + A060831(n) - 2*n = 2*A006218(n) - A006218(floor(n/2)) - 2*n with A006218(0) = 0, A006218(n) = sum(k=1, n, tau(k)) and now, by Dirichlet's asymptotic expression A006218(n) = n*log(n) + n*(2*gamma-1) + O(n^theta) (gamma = 0.57721..; 1/4 <= theta < 1/2), we have sum(k=1, n, a(k)) = 2*n*log(n) - (n/2)*log(n) + o(n*log(n)) = 1.5*n*log(n) + o(n*log(n)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 13 2002
a(n) = tau(2*n) - 2. - Michael Somos, Aug 30 2012
Sum_{k=1..n} a(k) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 7), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019

A338891 a(n) is the least number k such that the average number of odd divisors of {1..k} is >= n.

Original entry on oeis.org

1, 21, 165, 1274, 9435, 69720, 515230, 3807265, 28132035, 207869515, 1535959665, 11349295155, 83860579775

Offset: 1

Ilya Gutkovskiy, Nov 14 2020

			a(5) = 9435 because the average number of odd divisors of {1..9435} is >= 5.

Eric Weisstein's World of Mathematics, Odd Divisor Function.

Cf. A001227, A060831, A072334, A085829, A328331, A338943.

m = 1; sum = 0; s = {}; Do[sum += DivisorSigma[0, k/2^IntegerExponent[k, 2]]; If[sum >= m*k, AppendTo[s, k]; m++], {k, 1, 10^6}]; s (* Amiram Eldar, Nov 15 2020 *)

a(n) = my(s=1, k=1); while(s>valuation(k, 2))); k; \\ Michel Marcus, Nov 14 2020

a(n+1)/a(n) approaches e^2.

a(11)-a(12) from Amiram Eldar, Nov 16 2020

a(13) from Bill McEachen, Sep 01 2025

A075997 a(n) = [n/2] - [n/3] + [n/4] - [n/5] + [n/6] - ..., where [n/k] = floor(n/k).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 1, 4, 2, 3, 2, 5, 4, 5, 2, 6, 5, 6, 5, 8, 5, 6, 5, 10, 8, 9, 6, 9, 8, 9, 8, 13, 10, 11, 8, 12, 11, 12, 9, 14, 13, 14, 13, 16, 11, 12, 11, 18, 16, 17, 14, 17, 16, 17, 14, 19, 16, 17, 16, 21, 20, 21, 16, 22, 19, 20, 19, 22, 19, 20, 19, 26, 25, 26, 21, 24, 21, 22, 21

Offset: 0

Clark Kimberling, Sep 28 2002

a(n) is the number of terms among {floor(n/k)}, 1<=k<=n, which are even. - Leroy Quet, Jan 19 2006

			a(6) = [6/2]-[6/3]+[6/4]-[6/5]+[6/6]-[6/7]+... = 3-2+1-1+1-0+... = 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A000005, A006218, A060831, A059851.

A075997:=n->add(floor(n/(2*i))-floor((n-i)/(2*i)), i=1..n): seq(A075997(n), n=0..100); # Wesley Ivan Hurt, Jan 30 2016

Table[Sum[Floor[n/(2 i)] - Floor[(n - i)/(2 i)], {i, n}], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 30 2016 *)

a(n) = sum(k=2, n, (-1)^k*(n\k)); \\ Michel Marcus, Dec 20 2020

from math import isqrt
def A075997(n): return n+(s:=isqrt(n))**2-((t:=isqrt(m:=n>>1))**2<<1)-(sum(n//k for k in range(1,s+1))-(sum(m//k for k in range(1,t+1))<<1)<<1) # Chai Wah Wu, Oct 23 2023

a(n) = n - A059851(n).
a(n) = n - A006218(n) + 2*A006218(floor(n/2)). - Vladeta Jovovic, Oct 02 2002
a(n) = n - Sum_{n/2A000005(k). - Leroy Quet, Jan 19 2006
G.f.: ( Sum_{i>0} x^(2*i)/(1+x^i) )/(1-x). - Vladeta Jovovic, Apr 24 2006
a(n) = Sum_{i=1..n} floor(n/(2*i)) - floor((n-i)/(2*i)). - Wesley Ivan Hurt, Jan 30 2016
Conjecture: Let f(a,b)=1, if (a+b) mod |a-b| != (a mod |a-b|)+(b mod |a-b|), and 0 otherwise. a(n) = Sum_{k=1..n-1} f(n+k,n-k). - Benedict W. J. Irwin, Sep 23 2016
a(n) = Sum_{k=1..n} (floor((n-i)/i) mod 2 ). - Wesley Ivan Hurt, Dec 20 2020
a(n) ~ (1 - log(2))*n. - Vaclav Kotesovec, Jun 14 2025

A285902 Total number of partitions of all positive integers <= n into an even number of consecutive parts.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 14, 15, 16, 16, 17, 18, 20, 20, 21, 22, 23, 23, 25, 26, 27, 28, 29, 30, 32, 32, 33, 34, 35, 36, 38, 39, 40, 40, 41, 42, 44, 45, 46, 47, 49, 49, 51, 52, 53, 54, 55, 56, 58, 58, 60, 61, 62, 63, 65, 66, 67, 67, 68, 69, 72, 73, 74, 76, 77, 77, 79, 80, 81, 82

Offset: 1

Omar E. Pol, May 02 2017

nonn

a(n) is also the total number of odd divisors of k greater than sqrt(2*k), for k = 1..n.
Conjecture: a(n) is also the total number of pairs of equidistant subparts of all symmetric representations of sigma of all positive integers <= n.
For more information about the "subparts" of the symmetric representation of sigma see A279387 and A237593.

Partial sums of A131576.

Cf. A001227, A060831, A082647, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A237270, A279387, A245092, A262626, A285901.

a(n) = A060831(n) - A285901(n).

A340847 a(n) is the number of vertices in the diagram of the symmetric representation of sigma(n) with subparts.

Original entry on oeis.org

4, 6, 7, 10, 9, 13, 11, 14, 14, 15, 13, 23, 13, 17, 21, 22, 15, 26, 15, 25, 23, 21, 27, 35, 22, 21, 25, 29, 19, 41, 19, 30

Offset: 1

Omar E. Pol, Jan 24 2021

Theorem: Indices of even terms give A028982. Indices of odd terms give A028983.
If A001227(n) is odd then a(n) is even.
If A001227(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.
For another version see A340833 from which first differs at a(6).
For the definition of subparts see A279387. For more information about the subparts see also A237271, A280850, A280851, A296508, A335616.
Note that in this version of the diagram of the symmetric representation of sigma(n) all regions are called "subparts". The number of subparts equals A001227(n).

			Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             | |_|_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      6        7         10           9              13
.
For n = 6 the diagram has 13 vertices so a(6) = 13.
On the other hand the diagram has 14 edges and two subparts or regions, so applying Euler's formula we have that a(6) = 14 - 2 + 1 = 13.
.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
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 subparts 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  13             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  11               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  14                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  14                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | |
  10  15                     |_ _ _ _ _ _| |  _|  _| |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |  _|  _| |  _ _ _| | | | | |
  11  13                       |_ _ _ _ _ _| | |_ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  23                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |  _ _| |  _ _ _|
  13  13                           |_ _ _ _ _ _ _| | |  _| |_|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  17                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  21                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  22                                 |_ _ _ _ _ _ _ _ _|
...

Cf. A001227 (number of subparts or regions).
Cf. A340848 (number of edges).
Cf. A340833 (numer of vertices in the diagram only with parts).
Cf. A317293 (total number of vertices in the unified diagram).
Cf. A000203, A028982, A028983, A060831, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239660, A245092, A262626, A279387, A280850, A280851, A296508, A335616, A340846.

a(n) = A340848(n) - A001227(n) + 1 (Euler's formula).

A350122 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/(2*j-1))^k.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 8, 10, 5, 1, 16, 28, 17, 7, 1, 32, 82, 65, 27, 9, 1, 64, 244, 257, 127, 41, 11, 1, 128, 730, 1025, 627, 225, 55, 12, 1, 256, 2188, 4097, 3127, 1313, 353, 70, 15, 1, 512, 6562, 16385, 15627, 7809, 2419, 522, 93, 17, 1, 1024, 19684, 65537, 78127, 46721, 16841, 4114, 759, 115, 19

Offset: 1

Seiichi Manyama, Dec 16 2021

			Square array begins:
   1,  1,   1,    1,     1,      1,      1, ...
   2,  4,   8,   16,    32,     64,    128, ...
   4, 10,  28,   82,   244,    730,   2188, ...
   5, 17,  65,  257,  1025,   4097,  16385, ...
   7, 27, 127,  627,  3127,  15627,  78127, ...
   9, 41, 225, 1313,  7809,  46721, 280065, ...
  11, 55, 353, 2419, 16841, 117715, 823673, ...

Columns k=1..3 give A060831, A350143, A350144.
T(n,n) gives A350145.
Cf. A344725.

T[n_, k_] := Sum[Floor[n/(2*j - 1)]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 17 2021 *)

PARI
```
T(n, k) = sum(j=1, n, (n\(2*j-1))^k);
```

T(n, k) = sum(j=1, n, sumdiv(j, d, j/d%2*(d^k-(d-1)^k)));

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^(2*j)).

T(n,k) = Sum_{j=1..n} Sum_{d|j, j/d odd} d^k - (d - 1)^k.

A168508 Triangle read by rows: A101688 * A051731.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 27 2009

More precisely, form the product of the lower triangular matrix T defined in A101688 and the lower triangular matrix T defined in A051731. - N. J. A. Sloane, Dec 05 2020
Row sums = A060831: (1, 2, 4, 5, 7, 9, 11, 12, 15,...).
A168509 = A051731 * A101688

			First few rows of the triangle =
1;
1, 1;
2, 1, 1;
2, 1, 1, 1;
3, 1, 1, 1, 1;
3, 2, 1, 1, 1, 1;
4, 2, 1, 1, 1, 1, 1;
4, 2, 1, 1, 1, 1, 1, 1;
5, 2, 2, 1, 1, 1, 1, 1, 1;
5, 3, 2, 1, 1, 1, 1, 1, 1, 1;
6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1;
6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
7, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
8, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
8, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
9, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...

Cf. A051731, A060831, A101688, A168509.

a(29) = 1 inserted and more terms from Georg Fischer, May 29 2023

cp's OEIS Frontend

A285901 Total number of partitions of all positive integers <= n into an odd number of consecutive parts.

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A285900 Sum of all parts of all partitions of all positive integers <= n into consecutive parts.

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A263086 Partial sums of A099777, where A099777(n) gives the number of divisors of n-th even number.

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A069930 Number of integers of the form (n+k)/(n-k) with 1 <= k <= n-1.

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A338891 a(n) is the least number k such that the average number of odd divisors of {1..k} is >= n.

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A075997 a(n) = [n/2] - [n/3] + [n/4] - [n/5] + [n/6] - ..., where [n/k] = floor(n/k).

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A285902 Total number of partitions of all positive integers <= n into an even number of consecutive parts.

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A340847 a(n) is the number of vertices in the diagram of the symmetric representation of sigma(n) with subparts.

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