A067538 -id:A067538 - cp's OEIS frontend

A316413 Heinz numbers of integer partitions whose length divides their sum.

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 78, 79, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 94, 97, 98, 99, 100, 101, 103, 105, 107, 109, 110

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

In other words, partitions whose average is an integer.

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

			Sequence of partitions whose length divides their sum begins (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (1111), (7), (8), (42), (51), (9), (33), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1327 terms from R. J. Mathar)

Cf. A056239, A067538, A074761, A143773, A237984, A289508, A289509, A290103, A296150, A298423, A316428, A316431.

isA326413 := proc(n)
    psigsu := A056239(n) ;
    psigle := numtheory[bigomega](n) ;
    if modp(psigsu,psigle) = 0 then
        true;
    else
        false;
    end if;
end proc:
n := 1:
for i from 2 to 3000 do
    if isA326413(i) then
        printf("%d %d\n",n,i);
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019
# second Maple program:
q:= n-> (l-> nops(l)>0 and irem(add(i, i=l), nops(l))=0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..110])[];  # Alois P. Heinz, Nov 19 2021

Select[Range[2,100],Divisible[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]],PrimeOmega[#]]&]

A025065 Number of palindromic partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296, 11732, 11732, 14742, 14742, 18460, 18460, 23025, 23025, 28629, 28629

Offset: 0

Clark Kimberling

nonn

That is, the number of partitions of n into parts which can be listed in palindromic order.
Alternatively, number of partitions of n into parts from the set {1,2,4,6,8,10,12,...}. - T. D. Noe, Aug 05 2005
Also, partial sums of A035363.
Also number of partitions of n with at most one part occurring an odd number of times. - Reinhard Zumkeller, Dec 18 2013
The first Mathematica program computes terms of A025065; the second computes the k palindromic partitions of user-chosen n. - Clark Kimberling, Jan 20 2014
a(n) is the number of partitions p of n+1 such that 2*max(p) > n+1. - Clark Kimberling, Apr 20 2014.
From Gus Wiseman, Nov 28 2018: (Start)
Also the number of integer partitions of n + 2 that are the vertex-degrees of some hypertree. For example, the a(6) = 7 partitions of 8 that are the vertex-degrees of some hypertree, together with a realizing hypertree are:
     (41111): {{1,2},{1,3},{1,4},{1,5}}
     (32111): {{1,2},{1,3},{1,4},{2,5}}
     (22211): {{1,2},{1,3},{2,4},{3,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
    (221111): {{1,2},{1,3},{2,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}
(End)
Conjecture: a(n) is the length of maximal initial segment of A308355(n-1) that is identical to row n of A128628, for n >= 2. - Clark Kimberling, May 24 2019
From Gus Wiseman, May 21 2021: (Start)
The Heinz numbers of palindromic partitions are given by A265640. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
Also the number of integer partitions of n with a part greater than or equal to n/2. This is equivalent to Clark Kimberling's final comment above. The Heinz numbers of these partitions are given by A344414. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)     (8)
       (11)  (21)  (22)   (32)   (33)    (43)    (44)
                   (31)   (41)   (42)    (52)    (53)
                   (211)  (311)  (51)    (61)    (62)
                                 (321)   (421)   (71)
                                 (411)   (511)   (422)
                                 (3111)  (4111)  (431)
                                                 (521)
                                                 (611)
                                                 (4211)
                                                 (5111)
                                                 (41111)
Also the number of integer partitions of n with at least n/2 parts. The Heinz numbers of these partitions are given by A344296. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)   (21)   (22)    (221)    (222)     (2221)     (2222)
       (11)  (111)  (31)    (311)    (321)     (3211)     (3221)
                    (211)   (2111)   (411)     (4111)     (3311)
                    (1111)  (11111)  (2211)    (22111)    (4211)
                                     (3111)    (31111)    (5111)
                                     (21111)   (211111)   (22211)
                                     (111111)  (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

			The partitions for the first few values of n are as follows:
n: partitions .......................... number
1: 1 ................................... 1
2: 2 11 ................................ 2
3: 3 111 ............................... 2
4: 4 22 121 1111 ....................... 4
5: 5 131 212 11111 ..................... 4
6: 6 141 33 222 1221 11211 111111 ...... 7
7: 7 151 313 11311 232 21112 1111111 ... 7
From _Reinhard Zumkeller_, Jan 23 2010: (Start)
Partitions into 1,2,4,6,... for the first values of n:
1: 1 ....................................... 1
2: 2 11 .................................... 2
3: 21 111 .................................. 2
4: 4 22 211 1111 ........................... 4
5: 41 221 2111 11111 ....................... 4
6: 6 42 4211 222 2211 21111 111111.......... 7
7: 61 421 42111 2221 22111 211111 1111111 .. 7. (End)

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 101 terms from Reinhard Zumkeller that were corrected by _Georg Fischer_, Jan 20 2019)

Cf. A172033, A004277. - Reinhard Zumkeller, Jan 23 2010
Cf. A004526, A030019, A056503, A147878, A320921, A322136.
The bisections are both A000070.
The ordered version (palindromic compositions) is A016116.
The complement is counted by A233771 and A210249.
The case of palindromic prime signature is A242414.
Palindromic partitions are ranked by A265640, with complement A229153.
The case of palindromic plane trees is A319436.
The multiplicative version (palindromic factorizations) is A344417.
A000569 counts graphical partitions.
A027187 counts partitions of even length, ranked by A028260.
A035363 counts partitions into even parts, ranked by A066207.
A058696 counts partitions of even numbers, ranked by A300061.
A110618 counts partitions with length <= half sum, ranked by A344291.
Cf. A000041, A067538, A143773, A209816, A338914, A338915, A340387, A344296, A344414, A344415, A344416.

a025065 = p (1:[2,4..]) where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011

import Data.List (group)
a025065 = length . filter (<= 1) .
                   map (sum . map ((`mod` 2) . length) . group) . ps 1
   where ps x 0 = [[]]
         ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 18 2013

Map[Length[Select[IntegerPartitions[#], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &]] &, Range[40]] (* Peter J. C. Moses, Jan 20 2014 *)
n = 8; Select[IntegerPartitions[n], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &] (* Peter J. C. Moses, Jan 20 2014 *)
CoefficientList[Series[1/((1 - x) Product[1 - x^(2 n), {n, 1, 50}]), {x, 0, 60}], x] (* Clark Kimberling, Mar 14 2014 *)

N=66; q='q+O('q^N); Vec( 1/((1-q)*eta(q^2)) ) \\ Joerg Arndt, Mar 11 2014

a(n) = A000070(A004526(n)). - Reinhard Zumkeller, Jan 23 2010
G.f.: 1/((1-q)*prod(n>=1, 1-q^(2*n))). [Joerg Arndt, Mar 11 2014]
a(2*k+2) = a(2*k) + A000041(k + 1). - David A. Corneth, May 29 2021
a(n) ~ exp(Pi*sqrt(n/3)) / (2*Pi*sqrt(n)). - Vaclav Kotesovec, Nov 16 2021

Edited by N. J. A. Sloane, Dec 29 2007

Prepended a(0)=1, added more terms, Joerg Arndt, Mar 11 2014

A359893 Triangle read by rows where T(n,k) is the number of integer partitions of n with median k, where k ranges from 1 to n in steps of 1/2.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 2, 0, 2, 0, 0, 0, 1, 3, 0, 1, 2, 0, 0, 0, 0, 1, 4, 1, 2, 0, 3, 0, 0, 0, 0, 0, 1, 6, 1, 3, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 8, 1, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 11, 2, 7, 1, 3, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Jan 21 2023

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			Triangle begins:
  1
  1  0  1
  1  1  0  0  1
  2  0  2  0  0  0  1
  3  0  1  2  0  0  0  0  1
  4  1  2  0  3  0  0  0  0  0  1
  6  1  3  0  1  3  0  0  0  0  0  0  1
  8  1  6  0  2  0  4  0  0  0  0  0  0  0  1
 11  2  7  1  3  0  1  4  0  0  0  0  0  0  0  0  1
 15  2 10  3  4  0  2  0  5  0  0  0  0  0  0  0  0  0  1
 20  3 13  3  7  0  3  0  1  5  0  0  0  0  0  0  0  0  0  0  1
 26  4 19  3 11  1  4  0  2  0  6  0  0  0  0  0  0  0  0  0  0  0  1
For example, row n = 8 counts the following partitions:
  611       4211  422    .  332  .  44  .  .  .  .  .  .  .  8
  5111            521       431     53
  32111           2222              62
  41111           3221              71
  221111          3311
  311111          22211
  2111111
  11111111

Row sums are A000041.
Row lengths are 2n-1 = A005408(n-1).
Column k=1 is A027336(n+1).
For mean instead of median we have A058398, see also A008284, A327482.
The mean statistic is ranked by A326567/A326568.
Omitting half-steps gives A359901.
The odd-length case is A359902.
The median statistic is ranked by A360005(n)/2.
First appearances of medians are ranked by A360006, A360007.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts partitions w/ integer mean, strict A102627, ranked by A316413.
A240219 counts partitions w/ the same mean as median, complement A359894.
Cf. A325347, A349156, A359889, A359895, A359906, A359907, A360008.

Table[Length[Select[IntegerPartitions[n], Median[#]==k&]],{n,1,10},{k,1,n,1/2}]

A359901 Triangle read by rows where T(n,k) is the number of integer partitions of n with median k = 1..n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 2, 0, 1, 3, 1, 0, 0, 1, 4, 2, 3, 0, 0, 1, 6, 3, 1, 0, 0, 0, 1, 8, 6, 2, 4, 0, 0, 0, 1, 11, 7, 3, 1, 0, 0, 0, 0, 1, 15, 10, 4, 2, 5, 0, 0, 0, 0, 1, 20, 13, 7, 3, 1, 0, 0, 0, 0, 0, 1, 26, 19, 11, 4, 2, 6, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Jan 21 2023

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			Triangle begins:
   1
   1  1
   1  0  1
   2  2  0  1
   3  1  0  0  1
   4  2  3  0  0  1
   6  3  1  0  0  0  1
   8  6  2  4  0  0  0  1
  11  7  3  1  0  0  0  0  1
  15 10  4  2  5  0  0  0  0  1
  20 13  7  3  1  0  0  0  0  0  1
  26 19 11  4  2  6  0  0  0  0  0  1
  35 24 14  5  3  1  0  0  0  0  0  0  1
  45 34 17  8  4  2  7  0  0  0  0  0  0  1
  58 42 23 12  5  3  1  0  0  0  0  0  0  0  1
For example, row n = 9 counts the following partitions:
  (7,1,1)              (5,2,2)      (3,3,3)  (4,4,1)  .  .  .  .  (9)
  (6,1,1,1)            (6,2,1)      (4,3,2)
  (3,3,1,1,1)          (3,2,2,2)    (5,3,1)
  (4,2,1,1,1)          (4,2,2,1)
  (5,1,1,1,1)          (4,3,1,1)
  (3,2,1,1,1,1)        (2,2,2,2,1)
  (4,1,1,1,1,1)        (3,2,2,1,1)
  (2,2,1,1,1,1,1)
  (3,1,1,1,1,1,1)
  (2,1,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1)

Column k=1 is A027336(n+1).
For mean instead of median we have A058398, see also A008284, A327482.
Row sums are A325347.
The mean statistic is ranked by A326567/A326568.
Including half-steps gives A359893.
The odd-length case is A359902.
The median statistic is ranked by A360005(n)/2.
First appearances of medians are ranked by A360006, A360007.
A000041 counts partitions, strict A000009.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts partitions w/ integer mean, strict A102627, ranks A316413.
A240219 counts partitions w/ the same mean as median, complement A359894.
Cf. A008289, A237984, A327472, A349156, A359889, A359907.

Table[Length[Select[IntegerPartitions[n],Median[#]==k&]],{n,15},{k,n}]

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

Original entry on oeis.org

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

A102627 Number of partitions of n into distinct parts in which the number of parts divides n.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 4, 4, 5, 1, 15, 1, 7, 14, 17, 1, 28, 1, 40, 28, 11, 1, 99, 31, 13, 49, 99, 1, 186, 1, 152, 76, 17, 208, 425, 1, 19, 109, 699, 1, 584, 1, 433, 823, 23, 1, 1625, 437, 1140, 193, 746, 1, 2003, 1748, 2749, 244, 29, 1, 7404, 1, 31, 4158, 3258, 3766, 6307, 1

Offset: 1

Vladeta Jovovic, Feb 01 2005

			From _Gus Wiseman_, Sep 24 2019: (Start)
The a(1) = 1 through a(12) = 15 strict integer partitions whose average is an integer (A = 10, B = 11, C = 12):
  (1)  (2)  (3)  (4)   (5)  (6)    (7)  (8)   (9)    (A)   (B)  (C)
                 (31)       (42)        (53)  (432)  (64)       (75)
                            (51)        (62)  (531)  (73)       (84)
                            (321)       (71)  (621)  (82)       (93)
                                                     (91)       (A2)
                                                                (B1)
                                                                (543)
                                                                (642)
                                                                (651)
                                                                (732)
                                                                (741)
                                                                (831)
                                                                (921)
                                                                (5421)
                                                                (6321)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

The BI-numbers of these partitions are given by A326669 (numbers whose binary indices have integer mean).
The non-strict case is A067538.
Strict partitions with integer geometric mean are A326625.
Strict partitions whose maximum divides their sum are A326850.
Cf. A018818, A033630, A316413, A326622, A326843, A326851.

a:= proc(m) option remember; local b; b:=
      proc(n, i, t) option remember; `if`(i*(i+1)/2Alois P. Heinz, Sep 25 2019

npdp[n_]:=Count[Select[IntegerPartitions[n],Length[#]==Length[ Union[ #]]&], ?(Divisible[n,Length[#]]&)]; Array[npdp,70] (* _Harvey P. Dale, Feb 12 2016 *)
a[m_] := a[m] = Module[{b}, b[n_, i_, t_] := b[n, i, t] = If[i(i+1)/2 < n, 0, If[n == 0, If[Mod[m, t] == 0, 1, 0], b[n, i - 1, t] + b[n - i, Min[n - i, i - 1], t + 1]]]; If[PrimeQ[m], 1, b[m, m, 0]]];
Array[a, 100] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

A327482 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with mean d = A027750(n, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 1, 1, 7, 1, 1, 7, 5, 1, 1, 1, 1, 11, 15, 12, 6, 1, 1, 1, 1, 15, 7, 1, 1, 30, 19, 1, 1, 22, 34, 8, 1, 1, 1, 1, 30, 58, 27, 9, 1, 1, 1, 1, 42, 84, 64, 10, 1, 1, 105, 37, 1, 1, 56, 11, 1

Offset: 1

Gus Wiseman, Sep 13 2019

			Triangle begins:
  1
  1  1
  1  1
  1  2  1
  1  1
  1  3  3  1
  1  1
  1  5  4  1
  1  7  1
  1  7  5  1
  1  1
  1 11 15 12  6  1
  1  1
  1 15  7  1
  1 30 19  1
  1 22 34  8  1

Row sums are A067538.
The version for subsets is A327481.
Cf. A027750, A066571, A237984, A240850, A327474, A327483, A327484.

Table[Length[Select[IntegerPartitions[n],Mean[#]==d&]],{n,20},{d,Divisors[n]}]

Name edited by Peter Munn, Mar 05 2025

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[#]]]&]

A359902 Triangle read by rows where T(n,k) is the number of odd-length integer partitions of n with median k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 0, 0, 0, 1, 4, 2, 1, 0, 0, 0, 1, 4, 3, 2, 0, 0, 0, 0, 1, 7, 4, 3, 1, 0, 0, 0, 0, 1, 8, 6, 3, 2, 0, 0, 0, 0, 0, 1, 12, 8, 4, 3, 1, 0, 0, 0, 0, 0, 1, 14, 11, 5, 4, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Jan 21 2023

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			Triangle begins:
  1
  0  1
  1  0  1
  1  0  0  1
  2  1  0  0  1
  2  2  0  0  0  1
  4  2  1  0  0  0  1
  4  3  2  0  0  0  0  1
  7  4  3  1  0  0  0  0  1
  8  6  3  2  0  0  0  0  0  1
 12  8  4  3  1  0  0  0  0  0  1
 14 11  5  4  2  0  0  0  0  0  0  1
 21 14  8  4  3  1  0  0  0  0  0  0  1
 24 20 10  5  4  2  0  0  0  0  0  0  0  1
 34 25 15  6  5  3  1  0  0  0  0  0  0  0  1
For example, row n = 9 counts the following partitions:
  (7,1,1)              (5,2,2)      (3,3,3)  (4,4,1)  .  .  .  .  (9)
  (3,3,1,1,1)          (6,2,1)      (4,3,2)
  (4,2,1,1,1)          (2,2,2,2,1)  (5,3,1)
  (5,1,1,1,1)          (3,2,2,1,1)
  (2,2,1,1,1,1,1)
  (3,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1)

Column k=1 is A002865(n-1).
Row sums are A027193 (odd-length ptns), strict A067659.
This is the odd-length case of A359901, with half-steps A359893.
The median statistic is ranked by A360005(n)/2.
First appearances of medians are ranked by A360006, A360007.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A067538 counts partitions w/ integer mean, strict A102627, ranked by A316413.
A240219 counts partitions w/ the same mean as median, complement A359894.
A325347 counts partitions w/ integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
Cf. A008289, A026424, A327472, A359889, A359895, A359906, A359907, A359910.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&Median[#]==k&]],{n,15},{k,n}]

A307683 Number of partitions of n having a non-integer median.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 1, 7, 5, 11, 8, 18, 17, 31, 28, 47, 51, 75, 81, 119, 134, 181, 206, 277, 323, 420, 488, 623, 737, 922, 1084, 1352, 1597, 1960, 2313, 2819, 3330, 4029, 4743, 5704, 6722, 8030, 9434, 11234, 13175, 15601, 18262, 21552, 25184, 29612, 34518

Offset: 1

Clark Kimberling, Apr 24 2019

This sequence and A325347 partition the partition numbers, A000041.

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). - Gus Wiseman, Mar 16 2023

			a(7) counts these 4 partitions: [6,1], [5,2], [4,3], [3,2,1,1].

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..180

The complement is counted by A325347, strict A359907.
For mean instead of median we have A349156, strict A361391.
These partitions have ranks A359912, complement A359908.
The strict case is A360952.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A359893/A359901/A359902 count partitions by median.
Cf. A000016, A051293, A067538, A082550, A240219, A240850, A316413, A326567/A326568, A327475, A359897, A360005.

Table[Count[IntegerPartitions[n], q_ /; !IntegerQ[Median[q]]], {n, 10}]

cp's OEIS Frontend

A316413 Heinz numbers of integer partitions whose length divides their sum.

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A025065 Number of palindromic partitions of n.

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A359893 Triangle read by rows where T(n,k) is the number of integer partitions of n with median k, where k ranges from 1 to n in steps of 1/2.

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A359901 Triangle read by rows where T(n,k) is the number of integer partitions of n with median k = 1..n.

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

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A102627 Number of partitions of n into distinct parts in which the number of parts divides n.

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A327482 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with mean d = A027750(n, k).

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

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