A327472 -id:A327472 - cp's OEIS frontend

A237824 Number of partitions of n such that 2*(least part) >= greatest part.

1, 2, 3, 4, 5, 7, 7, 10, 11, 13, 14, 19, 18, 23, 25, 29, 30, 38, 37, 46, 48, 54, 57, 70, 69, 80, 85, 97, 100, 118, 118, 137, 144, 159, 168, 193, 195, 220, 233, 259, 268, 303, 311, 348, 367, 399, 419, 469, 483, 532, 560, 610, 639, 704, 732, 801, 841, 908, 954

Offset: 1

Clark Kimberling, Feb 16 2014

nonn

By conjugation, also the number of integer partitions of n whose greatest part appears at a middle position, namely at k/2, (k+1)/2, or (k+2)/2 where k is the number of parts. These partitions have ranks A362622. - Gus Wiseman, May 14 2023

			a(6) = 7 counts these partitions:  6, 42, 33, 222, 2211, 21111, 111111.
From _Gus Wiseman_, May 14 2023: (Start)
The a(1) = 1 through a(8) = 10 partitions such that 2*(least part) >= greatest part:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (42)      (322)      (53)
                    (1111)  (2111)   (222)     (2221)     (332)
                            (11111)  (2211)    (22111)    (422)
                                     (21111)   (211111)   (2222)
                                     (111111)  (1111111)  (22211)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part appears at a middle position:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (2211)    (2221)     (332)
                                     (111111)  (1111111)  (2222)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..300 from John Tyler Rascoe)

Cf. A237821, A118096, A053263.
The complement is counted by A237820, ranks A362982.
For modes instead of middles we have A362619, counted by A171979.
These partitions have ranks A362981.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
Cf. A002865, A008284, A237984, A238478, A238479, A327472, A359893, A362612, A362622, A384426.

z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] == Max[p]], {n, z}] (* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* this sequence *)
(* or *)
nmax = 100; Rest[CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j,k,2*k}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 13 2025 *)
(* or *)
nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(2*k - 1))*(1 - x^(2*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax+1)]; s += x^k/(1 - x^k)/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 14 2025 *)

N=60; x='x+O('x^N);
gf = sum(m=1, N, (x^m)/(1-x^m)) + sum(i=1, N, sum(j=1, i, x^((2*i)+j)/prod(k=0, j, 1 - x^(k+i))));
Vec(gf) \\ John Tyler Rascoe, Mar 07 2024

G.f.: Sum_{m>0} x^m/(1-x^m) + Sum_{i>0} Sum_{j=1..i} x^((2*i)+j) / Product_{k=0..j} (1 - x^(k+i)). - John Tyler Rascoe, Mar 07 2024
G.f.: Sum_{k>=1} x^k / Product_{j=k..2*k} (1 - x^j). - Vaclav Kotesovec, Jun 13 2025
a(n) ~ phi^(3/2) * exp(Pi*sqrt(2*n/15)) / (5^(1/4) * sqrt(2*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 14 2025

A363723 Number of integer partitions of n having a unique mode equal to the mean, i.e., partitions whose mean appears more times than each of the other parts.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 60, 15, 18, 37, 60, 2, 129, 2, 104, 80, 35, 104, 352, 2, 49, 168, 501, 2, 556, 2, 489, 763, 92, 2, 1799, 292, 985, 649, 1296, 2, 2233, 1681, 3379, 1204, 225, 2, 10661

Offset: 0

Gus Wiseman, Jun 24 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(n) partitions for n = 6, 8, 12, 14, 16 (A..G = 10..16):
  (6)       (8)         (C)             (E)               (G)
  (33)      (44)        (66)            (77)              (88)
  (222)     (2222)      (444)           (2222222)         (4444)
  (111111)  (3221)      (3333)          (3222221)         (5443)
            (11111111)  (4332)          (3322211)         (6442)
                        (5331)          (4222211)         (7441)
                        (222222)        (11111111111111)  (22222222)
                        (322221)                          (32222221)
                        (422211)                          (33222211)
                        (111111111111)                    (42222211)
                                                          (52222111)
                                                          (1111111111111111)

Partitions containing their mean are counted by A237984, ranks A327473.
For median instead of mode we have A240219, ranks A359889.
Partitions missing their mean are counted by A327472, ranks A327476.
The case of non-constant partitions is A362562.
Including median also gives A363719, ranks A363727.
Allowing multiple modes gives A363724.
Requiring multiple modes gives A363731.
For median instead of mean we have A363740.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median.
A362608 counts partitions with a unique mode.
Cf. A325347, A326567/A326568, A363720, A363725, A363730.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],{Mean[#]}==modes[#]&]],{n,30}]

A348551 Heinz numbers of integer partitions whose mean is not an integer.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 20, 24, 26, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 80, 86, 92, 93, 95, 96, 102, 104, 106, 108, 112, 114, 117, 119, 120, 122, 123, 124, 126, 130, 132, 135, 136, 140, 141, 142

Offset: 1

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms and their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  42: {1,2,4}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}

A version counting nonempty subsets is A000079 - A051293.
A version counting factorizations is A001055 - A326622.
A version counting compositions is A011782 - A271654.
A version for prime factors is A175352, complement A078175.
A version for distinct prime factors A176587, complement A078174.
The complement is A316413, counted by A067538, strict A102627.
The geometric version is the complement of A326623.
The conjugate version is the complement of A326836.
These partitions are counted by A349156.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A018818 counts partitions into divisors, ranked by A326841.
A143773 counts partitions into multiples of the length, ranked by A316428.
A236634 counts unbalanced partitions.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement A237984.
Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.

q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..142])[];  # Alois P. Heinz, Nov 19 2021

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

A238479 Number of partitions of n whose median is not a part.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 5, 8, 10, 13, 18, 23, 30, 40, 50, 64, 83, 104, 131, 166, 206, 256, 320, 394, 485, 598, 730, 891, 1088, 1318, 1596, 1932, 2326, 2797, 3360, 4020, 4804, 5735, 6824, 8108, 9624, 11392, 13468, 15904, 18737, 22048, 25914, 30400, 35619, 41686

Offset: 1

Clark Kimberling, Feb 27 2014

Also, the number of partitions p of n such that (1/2)*max(p) is a part of p.

Also the number of even-length integer partitions of n with distinct middle parts. For example, the partition (4,3,2,1) has middle parts {2,3} so is counted under a(10), but (3,2,2,1) has middle parts {2,2} so is not counted under a(8). - Gus Wiseman, May 13 2023

			a(6) counts these partitions:  51, 42, 2211 which all have an even number of parts, and their medians 3, 3 and 1.5 are not present. Note that the partitions 33 and 3111, although having an even number of parts, are not included in the count of a(6), but instead in that of A238478(6), as their medians, 3 for the former and 1 for the latter, are present in those partitions.

A. Blecher and A. Knopfmacher, Fixed points and matching points in partitions, Ramanujan J. 58 (2022), 23-41.

The complement is A238478, ranks A362618.
For mean instead of median we have A327472, complement A237984.
These partitions have ranks A362617.
A000041 counts integer partitions, even-length A027187.
A325347 counts partitions with integer median, complement A307683.
A359893/A359901/A359902 count partitions by median.
A359908 ranks partitions with integer median, complement A359912.
Cf. A002865, A008284, A053263, A238480, A238481, A240219, A359907, A360686, A360687, A360952.

Table[Count[IntegerPartitions[n], p_ /; !MemberQ[p, Median[p]]], {n, 40}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[p]/2]], {n, 50}]

my(q='q+O('q^50)); concat([0,0], Vec(sum(n=1,17,q^(3*n)/prod(k=1,2*n,1-q^k)))) \\ David Radcliffe, Jun 25 2025

from sympy.utilities.iterables import partitions
def A238479(n): return sum(1 for p in partitions(n) if (m:=max(p,default=0))&1^1 and m>>1 in p) # Chai Wah Wu, Sep 21 2023

a(n) = A000041(n) - A238478(n).
For all n, A027187(n) >= a(n). [Because when a partition of n has an odd number of parts, then it is not counted by this sequence (cf. A238478) and also some of the partitions with an even number of parts might be excluded here. Cf. Examples.] - Antti Karttunen, Feb 27 2014
From Jeremy Lovejoy, Sep 29 2022: (Start)
G.f.: Sum_{n>=1} q^(3*n)/Product_{k=1..2*n} (1-q^k).
a(n) ~ Pi/(2^(17/4)*3^(3/4)*n^(5/4))*exp(Pi*sqrt(2*n/3)). Proved by Blecher and Knopfmacher. (End)
a(n) = A087897(2*n) = A035294(n) - A078408(n-1). - Mathew Englander, May 20 2023

A363724 Number of integer partitions of n whose mean is a mode, i.e., partitions whose mean appears at least as many times as each of the other parts.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 5, 6, 2, 15, 2, 8, 15, 17, 2, 30, 2, 43, 30, 15, 2, 112, 36, 21, 60, 119, 2, 251, 2, 201, 126, 41, 271, 655, 2, 57, 250, 1060, 2, 1099, 2, 844, 1508, 107, 2, 3484, 802, 2068, 900, 2136, 2, 4558, 3513, 7071, 1630, 259, 2, 20260

Offset: 1

Gus Wiseman, Jun 24 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(n) partitions for n = 6, 10, 12:
  (6)            (10)                   (12)
  (3,3)          (5,5)                  (6,6)
  (2,2,2)        (2,2,2,2,2)            (4,4,4)
  (3,2,1)        (3,2,2,2,1)            (5,4,3)
  (1,1,1,1,1,1)  (4,2,2,1,1)            (6,4,2)
                 (1,1,1,1,1,1,1,1,1,1)  (7,4,1)
                                        (3,3,3,3)
                                        (4,3,3,2)
                                        (5,3,3,1)
                                        (6,3,2,1)
                                        (2,2,2,2,2,2)
                                        (3,2,2,2,2,1)
                                        (3,3,2,2,1,1)
                                        (4,2,2,2,1,1)
                                        (1,1,1,1,1,1,1,1,1,1,1,1)

For parts instead of modes we have A237984, complement A327472.
The case of a unique mode is A363723, non-constant A362562.
The case of more than one mode is A363731.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A362608 counts partitions with a unique mode.
A363719 = all three averages equal, ranks A363727, non-constant A363728.
A363720 = all three averages different, ranks A363730, unique mode A363725.
Cf. A240219, A326567/A326568, A363726, A363740.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],MemberQ[modes[#],Mean[#]]&]],{n,30}]

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}]

A238478 Number of partitions of n whose median is a part.

Original entry on oeis.org

1, 2, 2, 4, 5, 8, 11, 17, 22, 32, 43, 59, 78, 105, 136, 181, 233, 302, 386, 496, 626, 796, 999, 1255, 1564, 1951, 2412, 2988, 3674, 4516, 5524, 6753, 8211, 9984, 12086, 14617, 17617, 21211, 25450, 30514, 36475, 43550, 51869, 61707, 73230, 86821, 102706

Offset: 1

Clark Kimberling, Feb 27 2014

Also the number of integer partitions of n with a unique middle part. This means that either the length is odd or the two middle parts are equal. For example, the partition (4,3,2,1) has middle parts {2,3} so is not counted under a(10), but (3,2,2,1) has middle parts {2,2} so is counted under a(8). - Gus Wiseman, May 13 2023

			a(6) counts these partitions:  6, 411, 33, 321, 3111, 222, 21111, 111111.

For mean instead of median we have A237984, ranks A327473.
The complement is counted by A238479, ranks A362617.
These partitions have ranks A362618.
A000041 counts integer partitions.
A325347 counts partitions with integer median, complement A307683.
A359893/A359901/A359902 count partitions by median.
A359908 ranks partitions with integer median, complement A359912.
Cf. A002865, A008284, A053263, A238480, A240219, A327472, A359907, A360686, A360687, A362608.

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

a(n) + A238479(n) = A000041(n).
For all n, a(n) >= A027193(n) (because when a partition of n has an odd number of parts, its median is simply the part at the middle). - Antti Karttunen, Feb 27 2014
a(n) = A078408(n-1) - A282893(n). - Mathew Englander, May 24 2023

A363719 Number of integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 58, 15, 18, 37, 60, 2, 123, 2, 98, 79, 35, 103, 332, 2, 49, 166, 451, 2, 515, 2, 473, 738, 92, 2, 1561, 277, 839, 631, 1234, 2, 2043, 1560, 2867, 1156, 225, 2, 9020

Offset: 1

Gus Wiseman, Jun 19 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
Without loss of generality, we may assume there is a unique middle-part (A238478).
Includes all constant partitions.

			The a(n) partitions for n = 1, 2, 4, 6, 8, 12, 14, 16 (A..G = 10..16):
  1  2   4     6       8         C             E               G
     11  22    33      44        66            77              88
         1111  222     2222      444           2222222         4444
               111111  3221      3333          3222221         5443
                       11111111  4332          3322211         6442
                                 5331          4222211         7441
                                 222222        11111111111111  22222222
                                 322221                        32222221
                                 422211                        33222211
                                 111111111111                  42222211
                                                               52222111
                                                               1^16

For unequal instead of equal: A363720, ranks A363730, unique mode A363725.
The odd-length case is A363721.
These partitions have ranks A363727, nonprime A363722.
The case of non-constant partitions is A363728, ranks A363729.
The version for factorizations is A363741, see A359909, A359910.
Just two statistics:
- (mean) = (median) gives A240219, also A359889, A359895, A359897, A359899.
- (mean) != (median) gives A359894, also A359890, A359896, A359898, A359900.
- (mean) = (mode) gives A363723, see A363724, A363731.
- (median) = (mode) gives A363740.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or negative mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A027193, A237984, A325347, A326567/A326568, A327472, A363726, A363742.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], {Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]

A237821 Number of partitions of n such that 2*(least part) <= greatest part.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 11, 16, 25, 35, 48, 68, 92, 123, 164, 216, 282, 367, 471, 604, 769, 975, 1225, 1542, 1924, 2395, 2968, 3669, 4514, 5547, 6781, 8280, 10071, 12229, 14796, 17881, 21537, 25902, 31066, 37206, 44443, 53021, 63098, 74995, 88946, 105350, 124533

Offset: 1

Clark Kimberling, Feb 16 2014

By conjugation, also the number of integer partitions of n with different median from maximum, ranks A362980. - Gus Wiseman, May 15 2023

			a(6) = 7 counts these partitions:  51, 42, 411, 321, 3111, 2211, 21111.
From _Gus Wiseman_, May 15 2023: (Start)
The a(3) = 1 through a(8) = 16 partitions wirth 2*(least part) <= greatest part:
  (21)  (31)   (41)    (42)     (52)
        (211)  (221)   (51)     (61)
               (311)   (321)    (331)
               (2111)  (411)    (421)
                       (2211)   (511)
                       (3111)   (2221)
                       (21111)  (3211)
                                (4111)
                                (22111)
                                (31111)
                                (211111)
The a(3) = 1 through a(8) = 16 partitions with different median from maximum:
  (21)  (31)   (32)    (42)     (43)
        (211)  (41)    (51)     (52)
               (311)   (321)    (61)
               (2111)  (411)    (322)
                       (2211)   (421)
                       (3111)   (511)
                       (21111)  (3211)
                                (4111)
                                (22111)
                                (31111)
                                (211111)
(End)

The complement is counted by A053263, ranks A081306.
These partitions have ranks A069900.
The case of equality is A118096.
For < instead of <= we have A237820, ranks A362982.
For >= instead of <= we have A237824, ranks A362981.
The conjugate partitions have ranks A362980.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
Cf. A002865, A008284, A171979, A237984, A238478, A238479, A327472, A359893, A362612, A362622.

z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] = = Max[p]], {n, z}](* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* A237824 *)

G.f.: Sum_{i>=1} Sum_{j>=0} x^(3*i+j) /Product_{k=i..2*i+j} (1-x^k). - Seiichi Manyama, May 27 2023

A359895 Number of odd-length integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 1, 5, 5, 2, 5, 2, 8, 18, 1, 2, 19, 2, 24, 41, 20, 2, 9, 44, 31, 94, 102, 2, 125, 2, 1, 206, 68, 365, 382, 2, 98, 433, 155, 2, 716, 2, 1162, 2332, 196, 2, 17, 1108, 563, 1665, 3287, 2, 3906, 5474, 2005, 3083, 509, 2, 9029

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

The length and median of such a partition are integers with product n.

			The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)  (9)
            (111)       (11111)  (222)  (1111111)       (333)
                                 (321)                  (432)
                                                        (531)
                                                        (111111111)
The a(15) = 18 partitions:
  (15)
  (5,5,5)
  (6,5,4)
  (7,5,3)
  (8,5,2)
  (9,5,1)
  (3,3,3,3,3)
  (4,3,3,3,2)
  (4,4,3,2,2)
  (4,4,3,3,1)
  (5,3,3,2,2)
  (5,3,3,3,1)
  (5,4,3,2,1)
  (5,5,3,1,1)
  (6,3,3,2,1)
  (6,4,3,1,1)
  (7,3,3,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

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

This is the odd-length case of A240219, complement A359894, strict A359897.
These partitions are ranked by A359891, complement A359892.
The complement is counted by A359896.
The strict case is A359899, complement A359900.
The version for factorizations is A359910.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
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. A008289, A316313, A327472, A327475, A327482, A359889, A359906.

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

\\ P(n, k, m) is g.f. for k parts of max size m.
P(n, k, m)={polcoef(1/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)+h); polcoef(P(r, h, m)*P(r, h, r), r))))} \\ Andrew Howroyd, Jan 21 2023

a(p) = 2 for prime p. - Andrew Howroyd, Jan 21 2023

cp's OEIS Frontend

A237824 Number of partitions of n such that 2*(least part) >= greatest part.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A363723 Number of integer partitions of n having a unique mode equal to the mean, i.e., partitions whose mean appears more times than each of the other parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A348551 Heinz numbers of integer partitions whose mean is not an integer.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A238479 Number of partitions of n whose median is not a part.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A363724 Number of integer partitions of n whose mean is a mode, i.e., partitions whose mean appears at least as many times as each of the other parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A238478 Number of partitions of n whose median is a part.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A363719 Number of integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs