A065795 -id:A065795 - cp's OEIS frontend

A237984 Number of partitions of n whose mean is a part.

1, 2, 2, 3, 2, 5, 2, 6, 5, 8, 2, 21, 2, 14, 22, 30, 2, 61, 2, 86, 67, 45, 2, 283, 66, 80, 197, 340, 2, 766, 2, 663, 543, 234, 703, 2532, 2, 388, 1395, 4029, 2, 4688, 2, 4476, 7032, 1005, 2, 17883, 2434, 9713, 7684, 14472, 2, 25348, 17562, 37829, 16786, 3721

Offset: 1

Clark Kimberling, Feb 27 2014

a(n) = 2 if and only if n is a prime.

			a(6) counts these partitions:  6, 33, 321, 222, 111111.
From _Gus Wiseman_, Sep 14 2019: (Start)
The a(1) = 1 through a(10) = 8 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      432        22222
                           321              3221      531        32221
                           111111           4211      111111111  33211
                                            11111111             42211
                                                                 52111
                                                                 1111111111
(End)

Cf. A238478.
The Heinz numbers of these partitions are A327473.
A similar sequence for subsets is A065795.
Dominated by A067538.
The strict case is A240850.
Partitions without their mean are A327472.
Cf. A000016, A316413, A324753, A325705, A327478, A327482.

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Mean[p]]], {n, 40}]

from sympy.utilities.iterables import partitions
def A237984(n): return sum(1 for s,p in partitions(n,size=True) if not n%s and n//s in p) # Chai Wah Wu, Sep 21 2023

a(n) = A000041(n) - A327472(n). - Gus Wiseman, Sep 14 2019

A327473 Heinz numbers of integer partitions whose mean A326567/A326568 is a part.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 90, 97, 101, 103, 105, 107, 109, 110, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181

Offset: 1

Gus Wiseman, Sep 13 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  30: {1,2,3}
  31: {11}
  32: {1,1,1,1,1}
  37: {12}

A subsequence of A316413.
Complement of A327476.
The enumeration of these partitions by sum is given by A237984.
Subsets whose mean is a part are A065795.
Numbers whose binary indices include their mean are A327478.
Cf. A000016, A056239, A067538, A112798, A240850, A325706, A326567/A326568.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MemberQ[primeMS[#],Mean[primeMS[#]]]&]

A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94, 172, 316, 586, 1096, 2048, 3856, 7286, 13798, 26216, 49940, 95326, 182362, 349536, 671092, 1290556, 2485534, 4793492, 9256396, 17895736, 34636834, 67108864, 130150588, 252645136, 490853416

Offset: 0

N. J. A. Sloane

Also a(n+1) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. E.g., for n=5 there are 6 such sequences.
Also a(n+1) = number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 0 (mod n+1) = size of Varshamov-Tenengolts code VT_0(n). E.g., |VT_0(5)| = 6 = a(6).
Number of binary necklaces with an odd number of zeros. - Joerg Arndt, Oct 26 2015
Also, number of subsets of {1,2,...,n-1} which sum to 0 modulo n (cf. A063776). - Max Alekseyev, Mar 26 2016
From Gus Wiseman, Sep 14 2019: (Start)
Also the number of subsets of {1..n} containing n whose mean is an element. For example, the a(1) = 1 through a(8) = 16 subsets are:
  1  2  3    4    5      6      7        8
        123  234  135    246    147      258
                  345    456    357      468
                  12345  1236   567      678
                         1456   2347     1348
                         23456  2567     1568
                                12467    3458
                                13457    3678
                                34567    12458
                                1234567  14578
                                         23578
                                         24568
                                         45678
                                         123468
                                         135678
                                         2345678
(End)
Number of self-dual binary necklaces with 2n beads (cf. A263768, A007147). - Bernd Mulansky, Apr 25 2023

			For n=3 the 2 output sequences are 000111000111... and 010101...
For n=5 the 4 output sequences are those with periodic parts {0000011111, 0001011101, 0010011011, 01}.
For n=6 there are 6 such sequences.

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
J. Hedetniemi and K. R. Hutson, Equilibrium of shortest path load in ring network, Congressus Numerant., 203 (2010), 75-95. See p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Diff. Eqs. 20 (1) (2008) 201, eq. (39)

Seiichi Manyama, Table of n, a(n) for n = 0..3334 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, Univ. Buenos Aires (Argentina 2023). See also arXiv:2308.16257 [math.CO], 2023. See references.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.
A. E. Brouwer, The Enumeration of Locally Transitive Tournaments, Math. Centr. Report ZW138, Amsterdam, 1980.
S. Butenko, P. Pardalos, I. Sergienko, V. P. Shylo and P. Stetsyuk, Estimating the size of correcting codes using extremal graph problems, Optimization, 227-243, Springer Optim. Appl., 32, Springer, New York, 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Sébastien Designolle, Tamás Vértesi, and Sebastian Pokutta, Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms, arXiv:2310.20677 [quant-ph], 2023.
T. M. A. Fink, Exact dynamics of the critical Kauffman model with connectivity one, arXiv:2302.05314 [cond-mat.stat-mech], 2023.
R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer. Math. Monthly, 113 (2006), 887-896.
A. A. Kulkarni, N. Kiyavash and R. Sreenivas, On the Varshamov-Tenengolts Construction on Binary Strings, 2013.
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations, Pacific J. Math., 110 (1984), 203-221.
R. Pries and C. Weir, The Ekedahl-Oort type of Jacobians of Hermitian curves, arXiv preprint arXiv:1302.6261 [math.NT], 2013.
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, Challenge Problems: Independent Sets in Graphs
Yan Bo Ti, Gabriel Verret, and Lukas Zobernig, Abelian Varieties with p-rank Zero, arXiv:2203.08401 [math.NT], 2022.
Antonio Vera López, Luis Martínez, Antonio Vera Pérez, Beatriz Vera Pérez, and Olga Basova, Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions, Linear Algebra Appl. 530, 414-444 (2017).
Index entries for sequences related to tournaments
Index entries for sequences related to necklaces
Index entries for sequences related to subset sums modulo m

The main diagonal of table A068009, the left edge of triangle A053633.
Cf. A000048, A000031, A000013, A053634, A182469.
Subsets whose mean is an element are A065795.
Dominated by A082550.
Partitions containing their mean are A237984.
Subsets containing n but not their mean are A327477.
Cf. A051293, A135342, A240850, A327471, A327478.

a000016 0 = 1
a000016 n = (`div` (2 * n)) $ sum $
   zipWith (*) (map a000010 oddDivs) (map ((2 ^) . (div n)) $ oddDivs)
   where oddDivs = a182469_row n
-- Reinhard Zumkeller, May 01 2012

A000016 := proc(n) local d, t; if n = 0 then return 1 else t := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t := t + NumberTheory:-Totient(d)* 2^(n/d)/(2*n) fi od; return t fi end:

a[0] = 1; a[n_] := Sum[Mod[k, 2] EulerPhi[k]*2^(n/k)/(2*n), {k, Divisors[n]}]; Table[a[n], {n, 0, 35}](* Jean-François Alcover, Feb 17 2012, after Pari *)

a(n)=if(n<1,n >= 0,sumdiv(n,k,(k%2)*eulerphi(k)*2^(n/k))/(2*n));

from sympy import totient, divisors
def A000016(n): return sum(totient(d)<>(~n&n-1).bit_length(),generator=True))//n if n else 1 # Chai Wah Wu, Feb 21 2023

a(n) = Sum_{odd d divides n} (phi(d)*2^(n/d))/(2*n), n>0.
a(n) = A063776(n)/2.
a(n) = 2^(n-1) - A327477(n). - Gus Wiseman, Sep 14 2019

More terms from Michael Somos, Dec 11 1999

A327475 Number of subsets of {1..n} whose mean is an integer, where {} has mean 0.

Original entry on oeis.org

1, 2, 3, 6, 9, 16, 27, 46, 77, 136, 239, 426, 769, 1400, 2571, 4762, 8857, 16568, 31139, 58734, 111165, 211044, 401695, 766418, 1465489, 2807672, 5388783, 10359850, 19946833, 38459624, 74251095, 143524762, 277742489, 538043664, 1043333935, 2025040766, 3933915349

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

			The a(0) = 1 through a(5) = 16 subsets:
  {}  {}   {}   {}       {}       {}
      {1}  {1}  {1}      {1}      {1}
           {2}  {2}      {2}      {2}
                {3}      {3}      {3}
                {1,3}    {4}      {4}
                {1,2,3}  {1,3}    {5}
                         {2,4}    {1,3}
                         {1,2,3}  {1,5}
                         {2,3,4}  {2,4}
                                  {3,5}
                                  {1,2,3}
                                  {1,3,5}
                                  {2,3,4}
                                  {3,4,5}
                                  {1,2,4,5}
                                  {1,2,3,4,5}

If the subset is required to contain n, we get A063776.

Cf. A000016, A051293, A065795, A082550, A135342, A327474, A327481.

with(numtheory):
b:= n-> add(2^(n/d)*phi(d), d=select(x-> x::odd, divisors(n)))/n:
a:= proc(n) option remember; `if`(n=0, 1, b(n)-1+a(n-1)) end:
seq(a(n), n=0..36);  # Alois P. Heinz, Jan 13 2024

Table[Length[Select[Subsets[Range[n]],#=={}||IntegerQ[Mean[#]]&]],{n,0,10}]

from sympy import totient, divisors
def A327475(n): return sum((sum(totient(d)<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))-n+1 # Chai Wah Wu, Feb 22 2023

a(n) = A051293(n) + 1.

A327481 Triangle read by rows where T(n,k) is the number of nonempty subsets of {1..n} with mean k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 7, 3, 1, 1, 3, 9, 9, 3, 1, 1, 3, 9, 19, 9, 3, 1, 1, 3, 9, 25, 25, 9, 3, 1, 1, 3, 9, 29, 51, 29, 9, 3, 1, 1, 3, 9, 31, 75, 75, 31, 9, 3, 1, 1, 3, 9, 31, 93, 151, 93, 31, 9, 3, 1, 1, 3, 9, 31, 105, 235, 235, 105, 31, 9, 3, 1

Offset: 1

Gus Wiseman, Sep 13 2019

All terms are odd.

			Triangle begins:
                         1
                       1   1
                     1   3   1
                   1   3   3   1
                 1   3   7   3   1
               1   3   9   9   3   1
             1   3   9  19   9   3   1
           1   3   9  25  25   9   3   1
         1   3   9  29  51  29   9   3   1
       1   3   9  31  75  75  31   9   3   1
     1   3   9  31  93 151  93  31   9   3   1
   1   3   9  31 105 235 235 105  31   9   3   1
The subsets counted in row n = 5:
  {1}  {2}      {3}          {4}      {5}
       {1,3}    {1,5}        {3,5}
       {1,2,3}  {2,4}        {3,4,5}
                {1,3,5}
                {2,3,4}
                {1,2,4,5}
                {1,2,3,4,5}

Row sums are A051293.
The sequence of rows converges to A066571.
The version for partitions is A327482.
Cf. A000016, A063776, A065795, A135342, A327474, A327475, A327483, A327484.

Table[Length[Select[Subsets[Range[n]],Mean[#]==k&]],{n,10},{k,n}]

A063776 Number of subsets of {1,2,...,n} which sum to 0 modulo n.

Original entry on oeis.org

2, 2, 4, 4, 8, 12, 20, 32, 60, 104, 188, 344, 632, 1172, 2192, 4096, 7712, 14572, 27596, 52432, 99880, 190652, 364724, 699072, 1342184, 2581112, 4971068, 9586984, 18512792, 35791472, 69273668, 134217728, 260301176, 505290272, 981706832

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 16 2001

From Gus Wiseman, Sep 14 2019: (Start)
Also the number of subsets of {1..n} that are empty or contain n and have integer mean. If the subsets are not required to contain n, we get A327475. For example, the a(1) = 2 through a(6) = 12 subsets are:
  {}   {}   {}       {}       {}           {}
  {1}  {2}  {3}      {4}      {5}          {6}
            {1,3}    {2,4}    {1,5}        {2,6}
            {1,2,3}  {2,3,4}  {3,5}        {4,6}
                              {1,3,5}      {1,2,6}
                              {3,4,5}      {1,5,6}
                              {1,2,4,5}    {2,4,6}
                              {1,2,3,4,5}  {4,5,6}
                                           {1,2,3,6}
                                           {1,4,5,6}
                                           {2,3,5,6}
                                           {2,3,4,5,6}
(End)

			G.f. = 2*x + 2*x^2 + 4*x^3 + 4*x^4 + 8*x^5 + 12*x^6 + 20*x^7 + 32*x^8 + 60*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..3333 (first 200 terms from T. D. Noe)
T. Amdeberhan, M. Can and V. Moll, Broken bracelets, Molien series, paraffin wax and the elliptic curve 48a4, SIAM Journal of Discrete Math., v.25, 2011, p. 1843. See equation (1.2).
Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
N. Kitchloo and L. Pachter, An interesting result about subset sums, preprint (pdf file), 1993.
N. Kitchloo and L. Pachter, An interesting result about subset sums, preprint (pdf file), 1993.

The superdiagonal of A068009.
Cf. A000010, A000013, A051293, A053633, A053634, A053636, A054539, A082550.
Cf. A000016, A065795, A327475, A327481.

a063776 n = a053636 n `div` n  -- Reinhard Zumkeller, Sep 13 2013

Table[a = Select[ Divisors[n], OddQ[ # ] &]; Apply[Plus, 2^(n/a)*EulerPhi[a]]/n, {n, 1, 35}]
a[ n_] := If[ n < 1, 0, 1/n Sum[ Mod[ d, 2] EulerPhi[ d] 2^(n / d), {d, Divisors[ n]}]]; (* Michael Somos, May 09 2013 *)
Table[Length[Select[Subsets[Range[n]],#=={}||MemberQ[#,n]&&IntegerQ[Mean[#]]&]],{n,0,10}] (* Gus Wiseman, Sep 14 2019 *)

{a(n) = if( n<1, 0, 1 / n * sumdiv( n, d, (d % 2) * eulerphi(d) * 2^(n / d)))}; /* Michael Somos, May 09 2013 */

a(n) = sumdiv(n, d, (d%2)* 2^(n/d)*eulerphi(d))/n; \\ Michel Marcus, Feb 10 2016

from sympy import totient, divisors
def A063776(n): return (sum(totient(d)<>(~n&n-1).bit_length(),generator=True))<<1)//n # Chai Wah Wu, Feb 21 2023

a(n) = (1/n) * Sum_{d divides n and d is odd} 2^(n/d) * phi(d).
a(n) = (1/n) * A053636(n). - Michael Somos, May 09 2013
a(n) = 2 * A000016(n).
For odd n, a(n) = A000031(n).
G.f.: -Sum_{m >= 0} (phi(2*m + 1)/(2*m + 1)) * log(1 - 2*x^(2*m + 1)). - Petros Hadjicostas, Jul 13 2019
a(n) = A082550(n) + 1. - Gus Wiseman, Sep 14 2019

More terms from Vladeta Jovovic, Aug 20 2001

A359897 Number of strict integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 4, 4, 7, 6, 6, 10, 7, 10, 13, 11, 9, 20, 10, 20, 18, 21, 12, 30, 24, 28, 27, 30, 15, 73, 16, 37, 43, 45, 67, 74, 19, 55, 71, 126, 21, 150, 22, 75, 225, 78, 24, 183, 126, 245, 192, 132, 27, 284, 244, 403, 303, 120, 30, 828

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(1) = 1 through a(9) = 7 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)  (5,4)
                          (4,1)  (5,1)    (5,2)  (6,2)  (6,3)
                                 (3,2,1)  (6,1)  (7,1)  (7,2)
                                                        (8,1)
                                                        (4,3,2)
                                                        (5,3,1)

The non-strict version is A240219, complement A359894, ranked by A359889.
The complement is counted by A359898.
The odd-length case is A359899, complement A359900.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A237984 counts partitions containing their mean, complement A327472.
A240850 counts strict partitions containing their mean, complement A240851.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
Cf. A065795, A066571, A067659, A082550, A102627, A135342, A316313, A327473, A327475, A328966, A359909.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Mean[#]==Median[#]&]],{n,0,30}]

A359899 Number of strict odd-length integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 6, 1, 1, 6, 1, 5, 7, 1, 1, 8, 12, 1, 9, 2, 1, 33, 1, 1, 11, 1, 50, 12, 1, 1, 13, 70, 1, 46, 1, 1, 122, 1, 1, 16, 102, 155, 17, 1, 1, 30, 216, 258, 19, 1, 1, 310, 1, 1, 666, 1, 382, 23, 1, 1, 23, 1596, 1, 393, 1, 1

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(30) = 33 partitions:
  (30)  (11,10,9)  (8,7,6,5,4)
        (12,10,8)  (9,7,6,5,3)
        (13,10,7)  (9,8,6,4,3)
        (14,10,6)  (9,8,6,5,2)
        (15,10,5)  (10,7,6,4,3)
        (16,10,4)  (10,7,6,5,2)
        (17,10,3)  (10,8,6,4,2)
        (18,10,2)  (10,8,6,5,1)
        (19,10,1)  (10,9,6,3,2)
                   (10,9,6,4,1)
                   (11,7,6,4,2)
                   (11,7,6,5,1)
                   (11,8,6,3,2)
                   (11,8,6,4,1)
                   (11,9,6,3,1)
                   (12,7,6,3,2)
                   (12,7,6,4,1)
                   (12,8,6,3,1)
                   (12,9,6,2,1)
                   (13,7,6,3,1)
                   (13,8,6,2,1)
                   (14,7,6,2,1)
                   (11,10,6,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Strict odd-length case of A240219, complement A359894, ranked by A359889.
Strict case of A359895, complement A359896, ranked by A359891.
Odd-length case of A359897, complement A359898.
The complement is counted by A359900.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000016, A065795, A066571, A240851, A359903, A359906, A359910.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]

\\ Q(n,k,m) is g.f. for k strict parts of max size m.
Q(n,k,m)={polcoef(prod(i=1, m, 1 + y*x^i + O(x*x^n)), k, y)}
a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)); if(r>=h*(h+1), polcoef(Q(r, h, m-1)*Q(r, h, r), r)))))} \\ Andrew Howroyd, Jan 21 2023

a(p) = 1 for prime p. - Andrew Howroyd, Jan 21 2023

A359900 Number of strict odd-length integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 5, 4, 8, 10, 8, 15, 18, 17, 26, 27, 31, 43, 51, 53, 59, 81, 87, 109, 127, 115, 169, 194, 213, 255, 243, 322, 379, 431, 478, 487, 629, 667, 804, 907, 902, 1151, 1294, 1439, 1530, 1674, 2031, 2290, 2559, 2829, 2973, 3296, 3939

Offset: 0

Gus Wiseman, Jan 21 2023

nonn

			The a(7) = 1 through a(16) = 15 partitions (A=10, B=11, C=12, D=13):
  (421)  (431)  (621)  (532)  (542)  (651)  (643)  (653)  (762)  (754)
         (521)         (541)  (632)  (732)  (652)  (743)  (843)  (763)
                       (631)  (641)  (831)  (742)  (752)  (861)  (853)
                       (721)  (731)  (921)  (751)  (761)  (942)  (862)
                              (821)         (832)  (842)  (A32)  (871)
                                            (841)  (851)  (A41)  (943)
                                            (931)  (932)  (B31)  (952)
                                            (A21)  (941)  (C21)  (961)
                                                   (A31)         (A42)
                                                   (B21)         (A51)
                                                                 (B32)
                                                                 (B41)
                                                                 (C31)
                                                                 (D21)
                                                                 (64321)

This is the strict case of A359896, complement A359895, ranked by A359892.
This is the odd-length case of A359898, complement A359897.
The complement is counted by A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A359893/A359901/A359902 count partitions by median, ranked by A360005.
Cf. A000016, A065795, A066571, A102627, A240850, A240851, A327475, A359894, A359906, A359907, A359910.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&&Mean[#]!=Median[#]&]],{n,0,30}]

A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 6, 5, 11, 12, 14, 21, 29, 26, 44, 44, 58, 68, 92, 92, 118, 137, 165, 192, 241, 223, 324, 353, 405, 467, 518, 594, 741, 809, 911, 987, 1239, 1276, 1588, 1741, 1823, 2226, 2566, 2727, 3138, 3413, 3905, 4450, 5093, 5434, 6134

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(7) = 1 through a(13) = 11 partitions:
  (4,2,1)  (4,3,1)  (6,2,1)  (5,3,2)  (5,4,2)    (6,5,1)    (6,4,3)
           (5,2,1)           (5,4,1)  (6,3,2)    (7,3,2)    (6,5,2)
                             (6,3,1)  (6,4,1)    (8,3,1)    (7,4,2)
                             (7,2,1)  (7,3,1)    (9,2,1)    (7,5,1)
                                      (8,2,1)    (6,3,2,1)  (8,3,2)
                                      (5,3,2,1)             (8,4,1)
                                                            (9,3,1)
                                                            (10,2,1)
                                                            (5,4,3,1)
                                                            (6,4,2,1)
                                                            (7,3,2,1)

The non-strict version is ranked by A359890, complement A359889.
The non-strict version is A359894, complement A240219.
The complement is counted by A359897.
The odd-length case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000016, A065795, A066571, A082550, A135342, A240851, A327475, A327482, A328966, A359906.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Mean[#]!=Median[#]&]],{n,0,30}]

cp's OEIS Frontend

A237984 Number of partitions of n whose mean is a part.

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A327473 Heinz numbers of integer partitions whose mean A326567/A326568 is a part.

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A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

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A327475 Number of subsets of {1..n} whose mean is an integer, where {} has mean 0.

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A327481 Triangle read by rows where T(n,k) is the number of nonempty subsets of {1..n} with mean k.

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A063776 Number of subsets of {1,2,...,n} which sum to 0 modulo n.

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A359897 Number of strict integer partitions of n whose parts have the same mean as median.

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