A116608 -id:A116608 - cp's OEIS frontend

A376340 Sorted positions of first appearances in A057820, the sequence of first differences of prime-powers.

1, 4, 9, 12, 18, 24, 34, 47, 60, 79, 117, 178, 198, 206, 215, 244, 311, 402, 465, 614, 782, 1078, 1109, 1234, 1890, 1939, 1961, 2256, 2290, 3149, 3377, 3460, 3502, 3722, 3871, 4604, 4694, 6634, 8073, 8131, 8793, 12370, 12661, 14482, 14990, 15912, 17140, 19166

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    34: {1,7}
    47: {15}
    60: {1,1,2,3}
    79: {22}
   117: {2,2,6}
   178: {1,24}
   198: {1,2,2,5}
   206: {1,27}
   215: {3,14}
   244: {1,1,18}

For compression instead of sorted firsts we have A376308.
For run-lengths instead of sorted firsts we have A376309.
For run-sums instead of sorted firsts we have A376310.
The version for squarefree numbers is the unsorted version of A376311.
The unsorted version is A376341.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A078147, A110969, A120430, A174965, A373948, A375706.

q=Differences[Select[Range[100],PrimePowerQ]];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A060177 Triangle of generalized sum of divisors function, read by rows.

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 5, 2, 1, 6, 4, 2, 11, 2, 5, 13, 4, 10, 17, 3, 1, 15, 22, 4, 2, 25, 27, 2, 5, 37, 29, 6, 10, 52, 37, 2, 20, 67, 44, 4, 1, 30, 97, 44, 4, 2, 52, 117, 55, 5, 5, 77, 154, 59, 2, 10, 117, 184, 68, 6, 20, 162, 235, 71, 2, 36, 227, 277, 81, 6, 1, 58, 309, 338

Offset: 1

N. J. A. Sloane, Mar 20 2001

Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).

			Triangle turned on its side begins:
  1  2  2  3  2  4  2  4  3  4  2  6 ...
        1  2  5  6 11 13 17 22 27 29 ...
                 1  2  5 10 15 25 37 ...
                             1  2  5 ...

Alois P. Heinz, Rows n = 1..500, flattened
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Diagonals give A000005, A002133, A002134, A365630, A365631. Cf. A060043, A060044.

Cf. A116608 (reflected rows). - Alois P. Heinz, Jan 29 2014

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(b(n, i-1) +x*add(b(n-i*j, i-1), j=1..n/i))))
    end:
T:= n->(p->seq(coeff(p, x, degree(p)-k), k=0..degree(p)-1))(b(n$2)):
seq(T(n), n=1..25);  # Alois P. Heinz, Jan 29 2014

Reverse /@ Table[Length /@ Split[ Sort[Map[Length, Split /@ IntegerPartitions[n], {1}]]], {n, 24}] (* Wouter Meeussen, Apr 21 2012, updated by Jean-François Alcover, Jan 29 2014 *)

from math import isqrt
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A060177_gen(): # generator of terms
    return (sum(1 for p in partitions(n) if len(p)==k) for n in count(1) for k in range(isqrt((n<<3)+1)-1>>1,0,-1))
A060177_list = list(islice(A060177_gen(),30)) # Chai Wah Wu, Sep 15 2023

T(n,k) = Partitions of n using only k types of piles. Also, Sum_{k=1..A003056(n)} T(n,k)*k = A000070(n). Also, Sum_{k=1..A003056(n)} T(n,k)*(k-1) = A058884(n). - Naohiro Nomoto, Jan 24 2002

G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(m_1+m_2+...+m_k)/((1-q^m_1)*(1-q^m_2)*...*(1-q^m_k)) = Sum_n T(n, k)*q^n.

More terms from Naohiro Nomoto, Jan 24 2002

A334440 Irregular triangle T(n,k) read by rows: row n lists numbers of distinct parts of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 1, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3

Offset: 0

Gus Wiseman, May 05 2020

The total number of parts, counting duplicates, is A036043. The version for reversed partitions is A103921.

			Triangle begins:
  0
  1
  1 1
  1 2 1
  1 1 2 2 1
  1 2 2 2 2 2 1
  1 1 2 2 1 3 2 2 2 2 1
  1 2 2 2 2 2 3 2 2 3 2 2 2 2 1
  1 1 2 2 2 2 2 3 3 2 1 3 2 3 2 2 3 2 2 2 2 1

Robert Price, Table of n, a(n) for n = 0..9295 (25 rows).
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The number of not necessarily distinct parts is A036043.
The version for reversed partitions is A103921.
Ignoring length (sum/lex) gives A103921 (also).
a(n) is the number of distinct elements in row n of A334301.
The maximum part of the same partition is A334441.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colex order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Partitions counted by sum and number of distinct parts are A116608.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A001221, A049085, A124734, A185974, A296774, A299755, A334028, A334302, A334433, A334434, A334435, A334438.

Join@@Table[Length/@Union/@Sort[IntegerPartitions[n]],{n,0,10}]

a(n) = A001221(A334433(n)).

A353836 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 1, 0, 0, 0, 2, 5, 0, 0, 0, 0, 5, 5, 1, 0, 0, 0, 0, 2, 12, 1, 0, 0, 0, 0, 0, 7, 12, 3, 0, 0, 0, 0, 0, 0, 3, 19, 8, 0, 0, 0, 0, 0, 0, 0, 5, 27, 9, 1, 0, 0, 0, 0, 0, 0, 0, 2, 33, 20, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, May 26 2022

The run-sums of a sequence are the sums of its maximal consecutive constant subsequences (runs). For example, the run-sums of (2,2,1,1,1,3,2,2) are (4,3,3,4).

			Triangle begins:
  1
  0  1
  0  2  0
  0  2  1  0
  0  4  1  0  0
  0  2  5  0  0  0
  0  5  5  1  0  0  0
  0  2 12  1  0  0  0  0
  0  7 12  3  0  0  0  0  0
  0  3 19  8  0  0  0  0  0  0
  0  5 27  9  1  0  0  0  0  0  0
  0  2 33 20  1  0  0  0  0  0  0  0
  0 13 28 34  2  0  0  0  0  0  0  0  0
  0  2 48 46  5  0  0  0  0  0  0  0  0  0
  0  5 65 51 14  0  0  0  0  0  0  0  0  0  0
  0  4 57 99 15  1  0  0  0  0  0  0  0  0  0  0
For example, row n = 8 counts the following partitions:
  (8)         (53)       (431)
  (44)        (62)       (521)
  (422)       (71)       (3221)
  (2222)      (332)
  (41111)     (611)
  (221111)    (3311)
  (11111111)  (4211)
              (5111)
              (22211)
              (32111)
              (311111)
              (2111111)

Row sums are A000041.
Counting distinct parts instead of run-sums gives A116608.
Column k = 1 is A304442, ranked by A353833 (nonprime A353834).
The rank statistic is A353835, weak A353861, for compositions A353849.
A275870 counts collapsible partitions, ranked by A300273.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353864 counts rucksack partitions, ranked by A353866.
A353865 counts perfect rucksack partitions, ranked by A353867.
Cf. A008284, A047966, A071625, A165413, A175413, A325280, A333755.

Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Split[#]]]==k&]],{n,0,15},{k,0,n}]

A361394 Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 11, 15, 20, 30, 38, 49, 65, 83, 108, 139, 178, 224, 286, 358, 437, 550, 684, 837, 1037, 1269, 1553, 1889, 2295, 2770, 3359, 4035, 4843, 5808, 6951, 8312, 9902, 11752, 13958, 16531, 19541, 23037, 27162, 31911, 37488, 43950, 51463, 60127, 70229

Offset: 0

Gus Wiseman, Mar 17 2023

nonn

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)   (4)    (5)     (6)     (7)
       (11)  (21)  (22)   (32)    (33)    (43)
                   (31)   (41)    (42)    (52)
                   (211)  (221)   (51)    (61)
                          (311)   (321)   (322)
                          (2111)  (411)   (331)
                                  (2211)  (421)
                                  (3111)  (511)
                                          (2221)
                                          (3211)
                                          (4111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The complement is counted by A360254, ranks A360558.
These partitions have ranks A361395.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, reverse A058398.
A067538 counts partitions with integer mean, strict A102627.
A116608 counts partitions by number of distinct parts.
Cf. A106529, A144300, A237363, A237832, A240219, A316413, A349156.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>=0, 1, 0),
     `if`(i<1, 0, add(b(n-i*j, i-1, t+`if`(j>0, 2, 0)-j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 19 2023

Table[Length[Select[IntegerPartitions[n],2*Length[Union[#]]>=Length[#]&]],{n,0,30}]

A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 1, 2, 4, 0, 10, 0, 8, 16, 10, 0, 31, 0, 44, 44, 20, 0, 92, 50, 28, 98, 154, 0, 266, 0, 154, 194, 48, 434, 712, 0, 60, 348, 910, 0, 1198, 0, 1120, 2138, 88, 0, 2428, 1300, 1680, 912, 2506, 0, 4808, 4800, 5968, 1372, 140, 0, 14820, 0, 160

Offset: 1

Gus Wiseman, Mar 29 2023

nonn

Also partitions satisfying (maximum) = 2*(mean).

These are partitions whose diagram has the same size as its complement (see example).

			The a(6) = 2 through a(12) = 10 partitions:
  (411)   .  (4211)  (621)     (5221)   .  (822)
  (3111)             (321111)  (5311)      (831)
                               (42211)     (6222)
                               (43111)     (6321)
                                           (6411)
                                           (422211)
                                           (432111)
                                           (441111)
                                           (32211111)
                                           (33111111)
The partition y = (6,4,1,1) has diagram:
  o o o o o o
  o o o o . .
  o . . . . .
  o . . . . .
Since the partition and its complement (shown in dots) have the same size, y is counted under a(12).

For minimum instead of mean we have A118096.
For length instead of mean we have A237753.
For median instead of mean we have A361849, ranks A361856.
This is the equal case of A361851, unequal case A361852.
The strict case is A361854.
These partitions have ranks A361855.
This is the equal case of A361906, unequal case A361907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
A268192 counts partitions by complement size, ranks A326844.
Cf. A111907, A116608, A188814, A237755, A237824, A237984, A240219, A326849, A327482, A349156, A359894.

Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#==2n&]],{n,30}]

A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 11, 16, 24, 36, 52, 73, 101, 135, 184, 244, 321, 418, 543, 694, 889, 1127, 1427, 1789, 2242, 2787, 3463, 4276, 5271, 6465, 7921, 9655, 11756, 14254, 17262, 20830, 25102, 30152, 36172, 43270, 51691, 61594, 73300, 87023, 103189, 122099, 144296, 170193, 200497

Offset: 0

Gus Wiseman, Feb 12 2022

nonn

			The a(4) = 1 through a(9) = 16 partitions:
  (211)  (221)  (411)    (322)    (332)      (441)
         (311)  (2211)   (331)    (422)      (522)
                (21111)  (511)    (611)      (711)
                         (3211)   (3221)     (3321)
                         (22111)  (3311)     (4221)
                         (31111)  (4211)     (4311)
                                  (22211)    (5211)
                                  (32111)    (22221)
                                  (41111)    (32211)
                                  (221111)   (33111)
                                  (2111111)  (42111)
                                             (51111)
                                             (222111)
                                             (321111)
                                             (2211111)
                                             (3111111)
For example, the partition x = (2,1,1,1,1) has the permutation (1,1,2,1,1), with runs (1,1), (2), (1,1), which are not all distinct, so x is counted under a(6).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A144300.
The version for normal multisets is A283353.
The Heinz numbers of these partitions are A351201.
The complement is counted by A351204.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions, ranked by A333489.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A000041, A035363, A047993, A116608, A238130 or A238279, A325545, A329746, A350842, A351003, A351004, A351291.

Table[Length[Select[IntegerPartitions[n],MemberQ[Permutations[#],_?(!UnsameQ@@Split[#]&)]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
from itertools import permutations, groupby
from collections import Counter
def A351203(n):
    c = 0
    for s, p in partitions(n,size=True):
        for q in permutations(Counter(p).elements(),s):
            if max(Counter(tuple(g) for k, g in groupby(q)).values(),default=0) > 1:
                c += 1
                break
    return c # Chai Wah Wu, Oct 16 2023

a(n) = A000041(n) - A351204(n). - Andrew Howroyd, Jan 27 2024

a(26) onwards from Andrew Howroyd, Jan 27 2024

A360242 Number of integer partitions of n where the parts do not have the same mean as the distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 11, 19, 25, 43, 49, 82, 103, 136, 183, 258, 314, 435, 524, 687, 892, 1150, 1378, 1788, 2241, 2773, 3399, 4308, 5142, 6501, 7834, 9600, 11726, 14099, 16949, 20876, 25042, 30032, 35732, 43322, 51037, 61650, 72807, 86319, 102983, 122163

Offset: 0

Gus Wiseman, Feb 04 2023

nonn

			The a(1) = 0 through a(9) = 19 partitions:
  .  .  .  (211)  (221)   (411)    (322)     (332)      (441)
                  (311)   (3111)   (331)     (422)      (522)
                  (2111)  (21111)  (511)     (611)      (711)
                                   (2221)    (4211)     (3222)
                                   (3211)    (5111)     (3321)
                                   (4111)    (22211)    (4221)
                                   (22111)   (32111)    (4311)
                                   (31111)   (41111)    (5211)
                                   (211111)  (221111)   (6111)
                                             (311111)   (22221)
                                             (2111111)  (32211)
                                                        (33111)
                                                        (42111)
                                                        (51111)
                                                        (321111)
                                                        (411111)
                                                        (2211111)
                                                        (3111111)
                                                        (21111111)
For example, the partition y = (32211) has mean 9/5 and distinct parts {1,2,3} with mean 2, so y is counted under a(9).

The complement for multiplicities instead of distinct parts is A360068.
The complement is counted by A360243, ranks A360247.
For median instead of mean we have A360244, complement A360245.
These partitions have ranks A360246.
Sum of A360250 and A360251, ranks A360252 and A360253.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A058398 counts partitions by mean, also A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A360071 counts partitions by number of parts and number of distinct parts.
A360241 counts partitions whose distinct parts have integer mean.
Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156.

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

A360243 Number of integer partitions of n where the parts have the same mean as the distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 11, 11, 17, 13, 28, 19, 32, 40, 48, 39, 71, 55, 103, 105, 110, 105, 197, 170, 195, 237, 319, 257, 462, 341, 515, 543, 584, 784, 1028, 761, 973, 1153, 1606, 1261, 2137, 1611, 2368, 2815, 2575, 2591, 4393, 3798, 4602, 4663, 5777, 5121

Offset: 0

Gus Wiseman, Feb 04 2023

nonn

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (11111)  (51)      (61)       (62)
                                     (222)     (421)      (71)
                                     (321)     (1111111)  (431)
                                     (2211)               (521)
                                     (111111)             (2222)
                                                          (3221)
                                                          (3311)
                                                          (11111111)

For multiplicities instead of distinct parts we have A360068.
The complement is counted by A360242, ranks A360246.
For median instead of mean we have A360245, complement A360244.
These partitions have ranks A360247.
Cf. A360250 and A360251, ranks A360252 and A360253.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A058398 counts partitions by mean, also A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A360071 counts partitions by number of parts and number of distinct parts.
A360241 counts partitions whose distinct parts have integer mean.
Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156, A360069.

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

A373947 Halved and run-compressed first differences of consecutive odd primes.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 29 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The odd primes begin:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, ...
with differences:
2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, ...
with run-compression:
2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, ...
which is 2*a(n).

Half of A037201(n+1).
Run-compression of A001223.
A003242, A114901, A116608, A116861, A238279, A240085, A333755, A373948, A373949, A373951, A373953, A373954.

First/@Split[Differences[Select[Range[3,100],PrimeQ]]]/2

a(n) = A037201(n+1)/2.

cp's OEIS Frontend

A376340 Sorted positions of first appearances in A057820, the sequence of first differences of prime-powers.

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A060177 Triangle of generalized sum of divisors function, read by rows.

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A334440 Irregular triangle T(n,k) read by rows: row n lists numbers of distinct parts of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order.

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A353836 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums.

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A361394 Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).

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Maple

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A361853 Number of integer partitions of n such that (length) * (maximum) = 2n.

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A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

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A360242 Number of integer partitions of n where the parts do not have the same mean as the distinct parts.

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