cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A364345 Number of integer partitions of n without any three parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 27, 34, 43, 54, 67, 83, 102, 122, 151, 182, 218, 258, 313, 366, 443, 513, 611, 713, 844, 975, 1149, 1325, 1554, 1780, 2079, 2381, 2761, 3145, 3647, 4134, 4767, 5408, 6200, 7014, 8035, 9048, 10320, 11639, 13207, 14836, 16850
Offset: 0

Views

Author

Gus Wiseman, Jul 20 2023

Keywords

Examples

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (41)     (51)      (52)       (53)
                    (1111)  (311)    (222)     (61)       (62)
                            (11111)  (411)     (322)      (71)
                                     (3111)    (331)      (332)
                                     (111111)  (511)      (611)
                                               (4111)     (2222)
                                               (31111)    (3311)
                                               (1111111)  (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

Crossrefs

For subsets of {1..n} instead of partitions we have A007865 (sum-free sets), differences A288728.
Without re-using parts we have A236912, complement A237113.
Allowing the sum of any number of parts gives A237667 (cf. A108917).
The complement is counted by A363225, strict A363226, for subsets A093971.
The strict case is A364346.
These partitions have ranks A364347, complement A364348.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A025065, A026905, A111133, A240861, A275972, A320347, A325862, A326083, A363260.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,30}]

A363226 Number of strict integer partitions of n containing some three possibly equal parts (a,b,c) such that a + b = c. A variation of sum-full strict partitions.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 2, 3, 5, 4, 6, 7, 11, 11, 16, 18, 26, 29, 34, 42, 51, 62, 72, 84, 101, 119, 142, 166, 191, 226, 262, 300, 354, 405, 467, 540, 623, 705, 807, 927, 1060, 1206, 1369, 1551, 1760, 1998, 2248, 2556, 2861, 3236, 3628, 4100, 4587, 5152, 5756
Offset: 0

Views

Author

Gus Wiseman, Jul 19 2023

Keywords

Comments

Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.

Examples

			The a(3) = 1 through a(15) = 11 partitions (A=10, B=11, C=12):
  21  .  .  42   421  431  63   532   542   84    643   653   A5
            321       521  432  541   632   642   742   743   843
                           621  631   821   651   841   752   942
                                721   5321  921   A21   761   C21
                                4321        5421  5431  842   6432
                                            6321  6421  B21   6531
                                                  7321  5432  7431
                                                        6431  7521
                                                        6521  8421
                                                        7421  9321
                                                        8321  54321

Crossrefs

For subsets of {1..n} we have A093971 (sum-full sets), complement A007865.
The non-strict version is A363225, ranks A364348 (complement A364347).
The complement is counted by A364346, non-strict A364345.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions not re-using parts, complement A237113.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A025065, A026905, A085489, A108917, A237667, A237668 A240861, A275972, A320347, A326083.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]!={}&]],{n,0,30}]
  • Python
    from itertools import combinations_with_replacement
from collections import Counter
from sympy.utilities.iterables import partitions
def A363226(n): return sum(1 for p in partitions(n) if max(p.values(),default=0)==1 and any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # Chai Wah Wu, Sep 20 2023

Extensions

a(31)-a(56) from Chai Wah Wu, Sep 20 2023

A363944 Mean of the multiset of prime indices of n, rounded up.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 2, 6, 3, 3, 1, 7, 2, 8, 2, 3, 3, 9, 2, 3, 4, 2, 2, 10, 2, 11, 1, 4, 4, 4, 2, 12, 5, 4, 2, 13, 3, 14, 3, 3, 5, 15, 2, 4, 3, 5, 3, 16, 2, 4, 2, 5, 6, 17, 2, 18, 6, 3, 1, 5, 3, 19, 3, 6, 3, 20, 2, 21, 7, 3, 4, 5, 3, 22, 2, 2, 7
Offset: 1

Views

Author

Gus Wiseman, Jun 30 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Extending the terminology introduced at A124944, this is the "high mean" of prime indices.

Examples

			The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 2.

Crossrefs

Positions of first appearances are 1 and A000040.
Positions of 1's are A000079(n>0).
Before rounding up we had A326567/A326568.
For mode instead of mean we have A363487, low A363486.
For median instead of mean we have A363942, triangle A124944.
Rounding down instead of up gives A363943, triangle A363945.
The triangle for this statistic (high mean) is A363946.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363947 ranks partitions with rounded mean 1, counted by A363948.
A363949 ranks partitions with low mean 1, counted by A025065.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A051293, A124943, A215366, A327473, A327476, A327482, A359889, A362611, A363723, A363724, A363727, A363951.

Programs

  • Mathematica
    prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
meanup[y_]:=If[Length[y]==0,0,Ceiling[Mean[y]]];
Table[meanup[prix[n]],{n,100}]

A363949 Numbers whose prime indices have mean 1 when rounded down.

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448, 480, 486, 504, 512, 528, 540, 560, 576, 600, 640
Offset: 1

Views

Author

Gus Wiseman, Jul 02 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Crossrefs

These partitions are counted by A025065.
Before rounding down we had A326567/A326568.
For mode instead of mean we have A360015, counted by A241131.
For median instead of mean we have A363488, counted by A027336.
Positions of 1's in A363943, triangle A363945.
For the usual rounding (not low or high) we have A363948, counted by A363947.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
For mean 2 instead of 1 we have A363950, counted by A026905 redoubled.
Cf. A051293, A327473, A327476, A344296, A359889, A360013, A363723, A363727, A363946, A363951.

Programs

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

Formula

a(n) = 2*A344296(n).

A363946 Triangle read by rows where T(n,k) is the number of integer partitions of n with high mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1, 0, 1, 3, 2, 0, 1, 0, 1, 6, 3, 0, 0, 1, 0, 1, 6, 4, 3, 0, 0, 1, 0, 1, 11, 5, 4, 0, 0, 0, 1, 0, 1, 11, 13, 0, 4, 0, 0, 0, 1, 0, 1, 18, 9, 8, 5, 0, 0, 0, 0, 1, 0, 1, 18, 21, 10, 0, 5, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Gus Wiseman, Jun 30 2023

Keywords

Comments

Extending the terminology of A124944, the "high mean" of a multiset is obtained by taking the mean and rounding up.

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  3  0  1
  0  1  3  2  0  1
  0  1  6  3  0  0  1
  0  1  6  4  3  0  0  1
  0  1 11  5  4  0  0  0  1
  0  1 11 13  0  4  0  0  0  1
  0  1 18  9  8  5  0  0  0  0  1
  0  1 18 21 10  0  5  0  0  0  0  1
  0  1 29 28 12  0  6  0  0  0  0  0  1
  0  1 29 32 18 14  0  6  0  0  0  0  0  1
  0  1 44 43 23 16  0  7  0  0  0  0  0  0  1
  0  1 44 77 27 19  0  0  7  0  0  0  0  0  0  1
Row n = 7 counts the following partitions:
  .  (1111111)  (4111)    (511)  (61)  .  .  (7)
                (3211)    (421)  (52)
                (31111)   (331)  (43)
                (2221)    (322)
                (22111)
                (211111)

Crossrefs

Row sums are A000041.
Column k = 2 is A026905 redoubled, ranks A363950.
For median instead of mean we have triangle A124944, low A124943.
For mode instead of mean we have rank stat A363486, high A363487.
For median instead of mean we have rank statistic A363942, low A363941.
The rank statistic for this triangle is A363944.
The version for low mean is A363945, rank statistic A363943.
For mode instead of mean we have triangle A363953, low A363952.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A002865, A025065, A237984, A327472, A327482, A344296, A362612, A363723, A363724, A363731, A363948.

Programs

  • Mathematica
    meanup[y_]:=If[Length[y]==0,0,Ceiling[Mean[y]]];
Table[Length[Select[IntegerPartitions[n],meanup[#]==k&]],{n,0,15},{k,0,n}]

A363945 Triangle read by rows where T(n,k) is the number of integer partitions of n with low mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 4, 2, 0, 0, 1, 0, 4, 3, 3, 0, 0, 1, 0, 7, 4, 3, 0, 0, 0, 1, 0, 7, 10, 0, 4, 0, 0, 0, 1, 0, 12, 6, 7, 4, 0, 0, 0, 0, 1, 0, 12, 16, 8, 0, 5, 0, 0, 0, 0, 1, 0, 19, 21, 10, 0, 5, 0, 0, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Jun 30 2023

Keywords

Comments

Extending the terminology of A124943, the "low mean" of a multiset is its mean rounded down.

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  2  2  0  1
  0  4  2  0  0  1
  0  4  3  3  0  0  1
  0  7  4  3  0  0  0  1
  0  7 10  0  4  0  0  0  1
  0 12  6  7  4  0  0  0  0  1
  0 12 16  8  0  5  0  0  0  0  1
  0 19 21 10  0  5  0  0  0  0  0  1
  0 19 24 15 12  0  6  0  0  0  0  0  1
  0 30 32 18 14  0  6  0  0  0  0  0  0  1
  0 30 58 23 16  0  0  7  0  0  0  0  0  0  1
  0 45 47 57  0 19  0  7  0  0  0  0  0  0  0  1
Row k = 8 counts the following partitions:
  .  (41111)     (611)   .  (71)  .  .  .  (8)
     (32111)     (521)      (62)
     (311111)    (5111)     (53)
     (22211)     (431)      (44)
     (221111)    (422)
     (2111111)   (4211)
     (11111111)  (332)
                 (3311)
                 (3221)
                 (2222)

Crossrefs

Row sums are A000041.
Column k = 1 is A025065, ranks A363949.
For median instead of mean we have triangle A124943, high A124944.
Column k = 2 is A363745.
For median instead of mean we have rank statistic A363941, high A363942.
The rank statistic for this triangle is A363943.
The high version is A363946, rank statistic A363944.
For mode instead of mean we have A363952, rank statistic A363486.
For high mode instead of mean we have A363953, rank statistic A363487.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A002865, A026905, A237984, A327472, A327482, A344296, A362612, A363723, A363724, A363731, A363951.

Programs

  • Mathematica
    meandown[y_]:=If[Length[y]==0,0,Floor[Mean[y]]];
Table[Length[Select[IntegerPartitions[n],meandown[#]==k&]],{n,0,15},{k,0,n}]

A363950 Numbers whose prime indices have rounded-up mean 2.

Original entry on oeis.org

3, 6, 9, 10, 12, 18, 20, 24, 27, 28, 30, 36, 40, 48, 54, 56, 60, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 264, 270, 280, 288, 300, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448
Offset: 1

Views

Author

Gus Wiseman, Jul 05 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their prime indices begin:
     3: {2}
     6: {1,2}
     9: {2,2}
    10: {1,3}
    12: {1,1,2}
    18: {1,2,2}
    20: {1,1,3}
    24: {1,1,1,2}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    36: {1,1,2,2}
    40: {1,1,1,3}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    56: {1,1,1,4}
    60: {1,1,2,3}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}
    81: {2,2,2,2}

Crossrefs

For mean 1 we have A000079 except 1.
Partitions of this type are counted by A026905 redoubled.
Equals the complement of A000079 in A344296.
Positions of 2's in A363944 (counted by column 2 of A363946).
For rounded mean 1 we have A363948, counted by A363947.
For rounded-down mean 1 we have A363949, counted by A025065.
The rounded-down or low version is A363954, counted by A363745.
A316413 ranks partitions with integer mean, counted by A067538.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
Cf. A327473, A327476, A359889, A360005, A360013, A360015, A363727, A363943.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Ceiling[Mean[prix[#]]]==2&]

A363948 Numbers whose prime indices have mean < 3/2.

Original entry on oeis.org

2, 4, 8, 12, 16, 24, 32, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 432, 448, 480, 512, 576, 640, 768, 864, 896, 960, 1024, 1152, 1280, 1536, 1728, 1792, 1920, 2048, 2304, 2560, 2592, 2688, 2816, 2880, 3072, 3200, 3456, 3584, 3840, 4096, 4608
Offset: 1

Views

Author

Gus Wiseman, Jul 02 2023

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The initial terms, prime indices, and means:
    2: {1} -> 1
    4: {1,1} -> 1
    8: {1,1,1} -> 1
   12: {1,1,2} -> 4/3
   16: {1,1,1,1} -> 1
   24: {1,1,1,2} -> 5/4
   32: {1,1,1,1,1} -> 1
   48: {1,1,1,1,2} -> 6/5
   64: {1,1,1,1,1,1} -> 1
   72: {1,1,1,2,2} -> 7/5
   80: {1,1,1,1,3} -> 7/5
   96: {1,1,1,1,1,2} -> 7/6

Crossrefs

These partitions are counted by A363947.
Prime indices have mean A326567/A326568.
For low mode we have A360015, high A360013.
Positions of 1's in A363489.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363949 ranks partitions with low mean 1, counted by A025065.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A051293, A124944, A327473, A327476, A327482, A359889, A363727, A363942, A363943, A363946, A363951.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]<3/2&]

A211999 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Aug 14 2012

Keywords

Comments

The sequence lists the partitions of all positive integers. Each row of the irregular array is a partition of j.
At stage 1, we start with 1.
At stage j > 1, we write the partitions of j using the following rules:
First we copy the last A000041(j-1) rows of the array in descending order, as a mirror image, starting with the row that contains the part of size j-1. At the end of each new row, we added a part of size 1.
Second, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, such that the last row (or partition of j) is j.
Note that the table can be partially folded. In this case we have a three-dimensional structure in which each column contains parts of the same size (see example). Also the table can be completely folded, therefore stacked parts have the same size.

Examples

			A table of partitions.
---------------------------------------------------------
.              Expanded       Geometric  Side view of the
Partitions     version        model      folded table
---------------------------------------------------------
1;             1;             |*|                /
---------------------------------------------------------
1,1;           1,1;           |o|*|              \
2;             . 2;           |* *|               \
---------------------------------------------------------
2,1;           . 2,1;         |o o|*|             /
1,1,1;         1,1,1;         |o|o|*|            /
3;             . . 3;         |* * *|           /
---------------------------------------------------------
3,1;           . . 3,1;       |o o o|*|         \
1,1,1,1;       1,1,1,1;       |o|o|o|*|          \
2,1,1;         . 2,1,1;       |o o|o|*|           \
2,2;           . 2,. 2;       |* *|* *|            \
4;             . . . 4;       |* * * *|             \
---------------------------------------------------------
4,1;           . . . 4,1;     |o o o o|*|           /
2,2,1;         . 2,. 2,1;     |o o|o o|*|          /
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|         /
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|        /
3,1,1;         . . 3,1,1;     |o o o|o|*|       /
3,2;           . . 3,. 2;     |* * *|* *|      /
5;             . . . . 5;     |* * * * *|     /
---------------------------------------------------------
5,1;           . . . . 5,1;   |o o o o o|*|   \
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|    \
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|     \
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|      \
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|       \
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|        \
4,1,1;         . . . 4,1,1;   |o o o o|o|*|         \
2,2,2;         . 2, .2,. 2;   |* *|* *|* *|          \
4,2;           . . . 4,. 2;   |* * * *|* *|           \
3,3;           . . 3,. . 3;   |* * *|* * *|            \
6;             . . . . . 6;   |* * * * * *|             \
---------------------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Links

Crossrefs

Rows sums give A036042, n>=1.
Other versions are A211983, A211984, A211989. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A363947 Number of integer partitions of n with mean < 3/2.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 4, 4, 4, 7, 7, 7, 12, 12, 12, 19, 19, 19, 30, 30, 30, 45, 45, 45, 67, 67, 67, 97, 97, 97, 139, 139, 139, 195, 195, 195, 272, 272, 272, 373, 373, 373, 508, 508, 508, 684, 684, 684, 915, 915, 915, 1212, 1212, 1212, 1597, 1597, 1597, 2087
Offset: 0

Views

Author

Gus Wiseman, Jul 02 2023

Keywords

Examples

			The partition y = (2,2,1) has mean 5/3, which is not less than 3/2, so y is not counted under 5.
The a(1) = 1 through a(8) = 4 partitions:
  (1)  (11)  (111)  (211)   (2111)   (21111)   (22111)    (221111)
                    (1111)  (11111)  (111111)  (31111)    (311111)
                                               (211111)   (2111111)
                                               (1111111)  (11111111)

Crossrefs

The high version is A000012 (all ones).
This is A000070 with each term repeated three times (see A025065 for two).
These partitions have ranks A363948.
The complement is counted by A364059.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A327482 counts partitions by integer mean.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A000041, A002865, A026905, A027336, A237984, A241131, A327472, A363724, A363745, A363943, A363949.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Round[Mean[#]]==1&]],{n,0,15}]
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