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.

Showing 1-5 of 5 results.

A362559 Number of integer partitions of n whose weighted sum is divisible by n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 7, 8, 11, 14, 14, 18, 25, 28, 26, 42, 47, 52, 73, 77, 100, 118, 122, 158, 188, 219, 266, 313, 367, 412, 489, 578, 698, 809, 914, 1094, 1268, 1472, 1677, 1948, 2305, 2656, 3072, 3527, 4081, 4665, 5342, 6225, 7119, 8150, 9408
Offset: 1

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
Also the number of n-multisets of positive integers that (1) have integer mean, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.
Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

Examples

			The weighted sum of y = (4,2,2,1) is 1*4+2*2+3*2+4*1 = 18, which is a multiple of 9, so y is counted under a(9).
The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)     (7)        (8)       (9)
            (111)       (11111)  (222)   (3211)     (3311)    (333)
                                 (3111)  (1111111)  (221111)  (4221)
                                                              (222111)
                                                              (111111111)

Crossrefs

For median instead of mean we have A362558.
The complement is counted by A362560.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A264034 counts partitions by weighted sum.
A304818 = weighted sum of prime indices, row-sums of A359361.
A318283 = weighted sum of reversed prime indices, row-sums of A358136.
Cf. A001227, A051293, A067538, A067539, A240219, A261079, A322439, A326622, A359893, A360068, A360069, A362051.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]

A362560 Number of integer partitions of n whose weighted sum is not divisible by n.

Original entry on oeis.org

0, 1, 1, 4, 5, 8, 12, 19, 25, 38, 51, 70, 93, 124, 162, 217, 279, 360, 462, 601, 750, 955, 1203, 1502, 1881, 2336, 2892, 3596, 4407, 5416, 6623, 8083, 9830, 11943, 14471, 17488, 21059, 25317, 30376, 36424, 43489, 51906, 61789, 73498, 87186, 103253, 122098
Offset: 1

Views

Author

Gus Wiseman, Apr 28 2023

Keywords

Comments

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

Examples

			The weighted sum of y = (3,3,1) is 1*3+2*3+3*1 = 12, which is not a multiple of 7, so y is counted under a(7).
The a(2) = 1 through a(7) = 12 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)
              (31)    (41)    (42)      (52)
              (211)   (221)   (51)      (61)
              (1111)  (311)   (321)     (322)
                      (2111)  (411)     (331)
                              (2211)    (421)
                              (21111)   (511)
                              (111111)  (2221)
                                        (4111)
                                        (22111)
                                        (31111)
                                        (211111)

Crossrefs

For median instead of mean we have A322439 aerated, complement A362558.
The complement is counted by A362559.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A264034 counts partitions by weighted sum.
A304818 = weighted sum of prime indices, row-sums of A359361.
A318283 = weighted sum of reversed prime indices, row-sums of A358136.
Cf. A001227, A051293, A067538, A240219, A261079, A326622, A349156, A360068, A360069, A360241, A362051.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!Divisible[Total[Accumulate[Reverse[#]]],n]&]],{n,30}]

A362051 Number of integer partitions of 2n without a nonempty initial consecutive subsequence summing to n.

Original entry on oeis.org

1, 1, 2, 6, 11, 27, 44, 93, 149, 271, 432, 744, 1109, 1849, 2764, 4287, 6328, 9673, 13853, 20717, 29343, 42609, 60100, 85893, 118475, 167453, 230080, 318654, 433763, 595921, 800878, 1090189, 1456095, 1957032, 2600199, 3465459, 4558785, 6041381, 7908681
Offset: 0

Views

Author

Gus Wiseman, Apr 24 2023

Keywords

Comments

Even bisection of A362558.
a(0) = 1; a(n) = A000041(2n) - A322439(n). - Alois P. Heinz, Apr 27 2023

Examples

			The a(1) = 1 through a(4) = 11 partitions:
  (2)  (4)   (6)     (8)
       (31)  (42)    (53)
             (51)    (62)
             (222)   (71)
             (411)   (332)
             (2211)  (521)
                     (611)
                     (3221)
                     (3311)
                     (5111)
                     (32111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(4).

Crossrefs

The version for compositions is A000302, bisection of A213173.
The complement is counted by A322439.
Even bisection of A362558.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with all equal run-sums.
A325347 counts partitions with integer median, complement A307683.
A353836 counts partitions by number of distinct run-sums.
A359893/A359901/A359902 count partitions by median.
Cf. A108917, A169942, A237363, A325676, A353864, A360254, A360672, A360675, A360686, A360952, A362560.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],!MemberQ[Accumulate[#],n]&]],{n,0,15}]

A363525 Number of integer partitions of n with weighted sum divisible by reverse-weighted sum.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 5, 5, 3, 10, 4, 7, 13, 10, 8, 29, 10, 18, 39, 20, 20, 70, 29, 40, 105, 65, 55, 166, 73, 132, 242, 141, 129, 476, 183, 248, 580, 487, 312, 984, 422, 868, 1345, 825, 724, 2709, 949, 1505, 2756, 2902, 1611, 4664, 2289, 4942, 5828, 4278
Offset: 1

Views

Author

Gus Wiseman, Jun 10 2023

Keywords

Comments

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.

Examples

			The partition (6,5,4,3,2,1,1,1,1) has weighted sum 80, reverse 160, so is counted under a(24).
The a(n) partitions for n = 1, 2, 4, 6, 9, 12, 14 (A..E = 10-14):
  1  2   4     6       9          C             E
     11  22    33      333        66            77
         1111  222     711        444           65111
               111111  6111       921           73211
                       111111111  3333          2222222
                                  7311          71111111
                                  63111         11111111111111
                                  222222
                                  621111
                                  111111111111

Crossrefs

The case of equality (and reciprocal version) is A000005.
The strict case is A363528.
A000041 counts integer partitions, strict A000009.
A053632 counts compositions by weighted sum, rank statistic A029931/A359042.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
A363526 = partitions with weighted sum 3n, ranks A363530, reverse A363531.
Cf. A000016, A008284, A067538, A222855, A222970, A358137, A359755, A362558, A362559, A362560, A363527.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[#]], Total[Accumulate[Reverse[#]]]]&]],{n,30}]

A363528 Number of strict integer partitions of n with weighted sum divisible by reverse-weighted sum.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 6, 2, 3, 9, 3, 4, 11, 4, 5, 16, 6, 8, 24, 8, 10, 31, 11, 14, 41, 18, 18, 59, 21, 27, 74, 30, 32, 100, 35, 43, 128, 54, 53, 173, 58, 78, 215, 81, 88, 294, 97, 123, 362, 150, 146, 469, 162, 221, 577
Offset: 1

Views

Author

Gus Wiseman, Jun 10 2023

Keywords

Comments

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.

Examples

			The a(n) partitions for n = 1, 12, 15, 21, 24, 26:
  (1)  (12)     (15)       (21)          (24)          (26)
       (9,2,1)  (11,3,1)   (15,5,1)      (17,6,1)      (11,8,4,2,1)
                (9,3,2,1)  (16,3,2)      (18,4,2)      (12,6,5,2,1)
                           (11,7,2,1)    (12,9,2,1)    (13,5,4,3,1)
                           (12,5,3,1)    (13,7,3,1)
                           (10,5,3,2,1)  (14,5,4,1)
                                         (15,4,3,2)
                                         (10,8,3,2,1)
                                         (11,6,4,2,1)

Crossrefs

The non-strict version is A363525.
A000041 counts integer partitions, strict A000009.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, row-sums of A359361.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
A320387 counts multisets by weighted sum, zero-based A359678.
A363526 counts partitions with weighted sum 3n, reverse A363531.
Cf. A008284, A053632, A067538, A222855, A222970, A358137, A359754, A359755, A362558, A362559, A362560.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Divisible[Total[Accumulate[#]],Total[Accumulate[Reverse[#]]]]&]],{n,30}]
Showing 1-5 of 5 results.

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