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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jan 20 2023

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,0,30}]
  • PARI
    \\ Q(n,k,m) is g.f. for k strict parts of max size m.
Q(n,k,m)={polcoef(prod(i=1, m, 1 + y*x^i + O(x*x^n)), k, y)}
a(n)={if(n==0, 0, sumdiv(n, d, if(d%2, my(m=n/d, h=d\2, r=n-m*(h+1)); if(r>=h*(h+1), polcoef(Q(r, h, m-1)*Q(r, h, r), r)))))} \\ Andrew Howroyd, Jan 21 2023

Formula

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

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

Views

Author

Gus Wiseman, Feb 04 2023

Keywords

Examples

			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).

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Feb 04 2023

Keywords

Examples

			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)

Crossrefs

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.

Programs

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

A361851 Number of integer partitions of n such that (length) * (maximum) <= 2*n.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 18, 23, 31, 37, 51, 58, 75, 96, 116, 126, 184, 193, 253, 307, 346, 402, 511, 615, 678, 792, 1045, 1088, 1386, 1419, 1826, 2181, 2293, 2779, 3568, 3659, 3984, 4867, 5885, 6407, 7732, 8124, 9400, 11683, 13025, 13269, 16216, 17774, 22016
Offset: 1

Views

Author

Gus Wiseman, Mar 28 2023

Keywords

Comments

Also partitions such that (maximum) <= 2*(mean).
These are partitions whose complement (see example) has size <= n.

Examples

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (411)     (421)
                                     (2211)    (2221)
                                     (3111)    (3211)
                                     (21111)   (22111)
                                     (111111)  (211111)
                                               (1111111)
The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 <= 2*7, so y is counted under a(7).
The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 is not <= 2*9, so y is not counted under a(9).
The partition y = (3,2,1,1) has diagram:
  o o o
  o o .
  o . .
  o . .
with complement of size 5, and 5 <= 7, so y is counted under a(7).

Crossrefs

For length instead of mean we have A237755.
For minimum instead of mean we have A237824.
For median instead of mean we have A361848.
The equal case for median is A361849, ranks A361856.
The unequal case is A361852, median A361858.
The equal case is A361853, ranks A361855.
Reversing the inequality gives 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.
Cf. A111907, A237984, A240219, A324521, A324562, A327482, A349156, A360068, A360071, A360241, A361394, A361859.

Programs

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

A326851 Number of strict integer partitions of n whose length and maximum both divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 5, 1, 6, 1, 1, 1, 16, 1, 1, 1, 12, 1, 33, 1, 15, 1, 1, 1, 60, 1, 1, 1, 51, 1, 81, 1, 31, 57, 1, 1, 216, 1, 55, 1, 45, 1, 230, 1, 223, 1, 1, 1, 800, 1, 1, 314, 273, 1, 607, 1, 81, 1, 315, 1, 2404, 1, 1, 319
Offset: 0

Views

Author

Gus Wiseman, Jul 26 2019

Keywords

Examples

			The a(6) = 2 through a(24) = 16 partitions (1 terms not shown):
  6       12        15          16        18      20           24
  3,2,1   6,4,2     5,4,3,2,1   8,4,3,1   9,5,4   10,5,3,2     12,7,5
          6,5,1                 8,5,2,1   9,6,3   10,5,4,1     12,8,4
          6,3,2,1                         9,7,2   10,6,3,1     12,9,3
                                          9,8,1   10,7,2,1     12,10,2
                                                  10,4,3,2,1   12,11,1
                                                               8,7,5,4
                                                               8,7,6,3
                                                               12,5,4,3
                                                               12,6,4,2
                                                               12,6,5,1
                                                               12,7,3,2
                                                               12,7,4,1
                                                               12,8,3,1
                                                               12,9,2,1
                                                               8,6,4,3,2,1

Links

Crossrefs

The non-strict case is A326843.
Cf. A018818, A047993, A067538, A326837, A326842, A326849, A326852.

Programs

  • Mathematica
    Table[If[n==0,1,Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[n,Max[#]]&&Divisible[n,Length[#]]&]]],{n,0,30}]

A359906 Number of integer partitions of n with integer mean and integer median.

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 10, 9, 14, 2, 39, 2, 24, 51, 49, 2, 109, 2, 170, 144, 69, 2, 455, 194, 116, 381, 668, 2, 1378, 2, 985, 956, 316, 2043, 4328, 2, 511, 2293, 6656, 2, 8634, 2, 8062, 14671, 1280, 2, 26228, 8035, 15991, 11614, 25055, 2, 47201, 39810, 65092
Offset: 1

Views

Author

Gus Wiseman, Jan 21 2023

Keywords

Comments

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

Examples

			The a(1) = 1 through a(9) = 9 partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    11111  33      1111111  44        333
              31           42               53        432
              1111         51               62        441
                           222              71        522
                           321              2222      531
                           411              3221      621
                           111111           3311      711
                                            5111      111111111
                                            11111111

Crossrefs

For just integer mean we have A067538, strict A102627, ranked by A316413.
For just integer median we have A325347, strict A359907, ranked by A359908.
These partitions are ranked by A360009.
A000041 counts partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A051293 counts subsets with integer mean, median A000975.
A326567/A326568 gives mean of prime indices.
A326622 counts factorizations with integer mean, strict A328966.
A359893/A359901/A359902 count partitions by median.
A360005(n)/2 gives median of prime indices.
Cf. A000016, A082550, A237984, A240219, A326669, A327475, A349156, A359894, A359897, A359905.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], IntegerQ[Mean[#]]&&IntegerQ[Median[#]]&]],{n,1,30}]

A326643 Number of subsets of {1..n} whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 30, 31, 32, 33, 34, 35, 41, 46, 47, 70, 71, 72, 73, 74, 102, 103, 104, 105, 143, 144, 145, 146, 151, 152, 153, 154, 155, 161, 162, 163, 244, 252, 280, 281, 282, 283, 409, 410, 416, 417, 418, 419
Offset: 0

Views

Author

Gus Wiseman, Jul 16 2019

Keywords

Examples

			The a(1) = 1 through a(12) = 16 subsets:
  {1}  {1}  {1}  {1}  {1}  {1}  {1}  {1}    {1}    {1}    {1}    {1}
       {2}  {2}  {2}  {2}  {2}  {2}  {2}    {2}    {2}    {2}    {2}
            {3}  {3}  {3}  {3}  {3}  {3}    {3}    {3}    {3}    {3}
                 {4}  {4}  {4}  {4}  {4}    {4}    {4}    {4}    {4}
                      {5}  {5}  {5}  {5}    {5}    {5}    {5}    {5}
                           {6}  {6}  {6}    {6}    {6}    {6}    {6}
                                {7}  {7}    {7}    {7}    {7}    {7}
                                     {8}    {8}    {8}    {8}    {8}
                                     {2,8}  {9}    {9}    {9}    {9}
                                            {1,9}  {10}   {10}   {10}
                                            {2,8}  {1,9}  {11}   {11}
                                                   {2,8}  {1,9}  {12}
                                                          {2,8}  {1,9}
                                                                 {2,8}
                                                                 {3,6,12}
                                                                 {3,4,9,12}

Links

Crossrefs

Partial sums of A326644.
Subsets whose geometric mean is an integer are A326027.
Subsets whose mean is an integer are A051293.
Partitions with integer mean and geometric mean are A326641.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A067538, A067539, A078175, A082553, A102627, A326623, A326625, A326645, A326646, A326647.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,0,10}]

Extensions

More terms from David Wasserman, Aug 03 2019

A326848 Heinz numbers of integer partitions of m >= 0 whose length times maximum is a multiple of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 78, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 171, 173, 179, 181
Offset: 1

Views

Author

Gus Wiseman, Jul 26 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The enumeration of these partitions by sum is given by A326849.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    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}
   28: {1,1,4}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Crossrefs

Cf. A001222, A047993, A056239, A061395, A067538, A112798, A316413, A326836, A326843, A326847, A326849, A326851.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],#==1||Divisible[Max[primeMS[#]]*Length[primeMS[#]],Total[primeMS[#]]]&]

A326850 Number of strict integer partitions of n whose maximum part divides n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 1, 10, 1, 10, 5, 12, 1, 23, 1, 18, 15, 23, 1, 49, 1, 34, 36, 38, 1, 106, 1, 54, 79, 81, 1, 189, 1, 124, 162, 104, 1, 412, 1, 145, 307, 289, 1, 608, 12, 437, 559, 256, 1, 1432, 1, 340, 981, 976, 79, 1730, 1
Offset: 0

Views

Author

Gus Wiseman, Jul 28 2019

Keywords

Examples

			The initial terms count the following partitions:
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   5: (5)
   6: (6)
   6: (3,2,1)
   7: (7)
   8: (8)
   8: (4,3,1)
   9: (9)
  10: (10)
  10: (5,4,1)
  10: (5,3,2)
  11: (11)
  12: (12)
  12: (6,5,1)
  12: (6,4,2)
  12: (6,3,2,1)
  13: (13)
  14: (14)
  14: (7,6,1)
  14: (7,5,2)
  14: (7,4,3)
  14: (7,4,2,1)
  15: (15)
  15: (5,4,3,2,1)

Links

Crossrefs

Positions of 1's appear to be A308168.
The non-strict case is given by A067538.
Cf. A018818, A033630, A067538, A102627, A200745, A316413, A326625, A326836, A326843, A326851.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[n,Max[#]]&]],{n,0,30}]

A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 2, 4, 5, 6, 6, 7, 8, 11, 10, 13, 17, 18, 21, 24, 27, 30, 35, 39, 46, 53, 61, 68, 79, 87, 97, 110, 123, 139, 157, 175, 196, 222, 247, 278, 312, 347, 385, 433, 476, 531, 586, 651, 720, 800, 883, 979, 1085, 1200, 1325, 1464, 1614, 1777
Offset: 1

Views

Author

Gus Wiseman, Feb 01 2021

Keywords

Examples

			The a(1) = 1 through a(16) = 10 partitions (A..G = 10..16):
  1  2  3   4  5   6    7   8   9    A     B    C    D    E     F      G
        21     41  42   43  62  63   64    65   84   85   86    87     A6
                   321  61      81   82    83   A2   A3   A4    A5     C4
                                621  631   A1   642  C1   C2    C3     E2
                                     4321  632  651  643  653   E1     943
                                           641  921  652  932   654    952
                                                     931  941   942    961
                                                          8321  951    C31
                                                                C21    8431
                                                                8421   8521
                                                                54321

Links

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict version is A168659 (A340609/A340610).
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A064173 counts partitions of positive/negative rank (A340787/A340788).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A096401 counts strict partition with length equal to minimum.
A102627 counts strict partitions with length dividing sum.
A326842 counts partitions whose length and parts all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340829 counts strict partitions with Heinz number divisible by sum.
A340830 counts strict partitions with all parts divisible by length.
Cf. A003114, A006141, A064174, A117409, A143773 (A316428), A200750, A326843 (A326837), A330950 (A324851).

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Max@@#,Length[#]]&]],{n,30}]
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