A359907 -id:A359907 - cp's OEIS frontend

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

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

A360245 Number of integer partitions of n where the parts have the same median as the distinct parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 11, 13, 19, 19, 35, 33, 48, 66, 78, 88, 124, 138, 183, 219, 252, 306, 388, 450, 527, 643, 780, 903, 1097, 1266, 1523, 1784, 2107, 2511, 2966, 3407, 4019, 4667, 5559, 6364, 7492, 8601, 10063, 11634, 13469, 15469, 17985, 20558, 23812

Offset: 0

Gus Wiseman, Feb 05 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).

			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 example, the partition y = (6,4,4,4,1,1) has median 4, and the distinct parts {1,4,6} also have median 4, so y is counted under a(20).

For mean instead of median: A360242, ranks A360247, complement A360243.
These partitions have ranks A360249.
The complement is A360244, ranks A360248.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
A240219 counts partitions with mean equal to median, ranks A359889.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median.
A359894 counts partitions with mean different from median, ranks A359890.
A360071 counts partitions by number of parts and number of distinct parts.
Cf. A000975, A027193, A067659, A326619/A326620, A326621, A359902, A360241, A360246, A360250, A360251.

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

A360254 Number of integer partitions of n with more adjacent equal parts than distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 4, 7, 10, 12, 18, 28, 36, 52, 68, 92, 119, 161, 204, 269, 355, 452, 571, 738, 921, 1167, 1457, 1829, 2270, 2834, 3483, 4314, 5300, 6502, 7932, 9665, 11735, 14263, 17227, 20807, 25042, 30137, 36099, 43264, 51646, 61608, 73291, 87146, 103296

Offset: 0

Gus Wiseman, Feb 20 2023

nonn

None of these partitions is strict.

Also the number of integer partitions of n which, after appending 0, have first differences of median 0.

			The a(3) = 1 through a(9) = 10 partitions:
  (111)  (1111)  (11111)  (222)     (22111)    (2222)      (333)
                          (21111)   (31111)    (22211)     (22221)
                          (111111)  (211111)   (41111)     (33111)
                                    (1111111)  (221111)    (51111)
                                               (311111)    (222111)
                                               (2111111)   (411111)
                                               (11111111)  (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)
For example, the partition y = (4,4,3,1,1,1,1) has 0-appended differences (0,1,2,0,0,0,0), with median 0, so y is counted under a(15).

The non-prepended version is A237363.
These partitions have ranks A360558.
For any integer median (not just 0) we have A360688.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A067538, A102627, A240219, A359894, A360071, A360244, A360555, A360556.

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

A231147 Array of coefficients of numerator polynomials of the rational function p(n, x + 1/x), where p(n,x) = (x^n - 1)/(x - 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 5, 4, 9, 4, 5, 1, 1, 1, 1, 6, 5, 14, 9, 14, 5, 6, 1, 1, 1, 1, 7, 6, 20, 14, 29, 14, 20, 6, 7, 1, 1, 1, 1, 8, 7, 27, 20, 49, 29, 49, 20, 27, 7, 8, 1, 1, 1, 1, 9, 8, 35, 27, 76, 49, 99, 49, 76, 27, 35, 8, 9

Offset: 1

Clark Kimberling, Nov 05 2013

From Gus Wiseman, Mar 19 2023: (Start)
Also appears to be the number of nonempty subsets of {1,...,n} with median k, where k ranges from 1 to n in steps of 1/2, and the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). For example, row n = 5 counts the following subsets:
{1}  {1,2}  {2}      {1,4}      {3}          {2,5}      {4}      {4,5}  {5}
            {1,3}    {2,3}      {1,5}        {3,4}      {3,5}
            {1,2,3}  {1,2,3,4}  {2,4}        {1,3,4,5}  {1,4,5}
            {1,2,4}  {1,2,3,5}  {1,3,4}      {2,3,4,5}  {2,4,5}
            {1,2,5}             {1,3,5}                 {3,4,5}
                                {2,3,4}
                                {2,3,5}
                                {1,2,4,5}
                                {1,2,3,4,5}
Central diagonals T(n,(n+1)/2) appear to be A100066 (bisection A006134).
For mean instead of median we have A327481.
For partitions instead of subsets we have A359893, full steps A359901.
Central diagonals T(n,n/2) are A361801 (bisection A079309).
(End)

			Triangle begins:
  1
  1  1  1
  1  1  3  1  1
  1  1  4  3  4  1  1
  1  1  5  4  9  4  5  1  1
  1  1  6  5 14  9 14  5  6  1  1
  1  1  7  6 20 14 29 14 20  6  7  1  1
  1  1  8  7 27 20 49 29 49 20 27  7  8  1  1
  1  1  9  8 35 27 76 49 99 49 76 27 35  8  9  1  1
First 3 polynomials: 1, 1 + x + x^2, 1 + x + 3*x^2 + x^3 + x^4

John Tyler Rascoe, Rows n = 1..100, flattened

Cf. A231148.
Row sums are 2^n-1 = A000225(n).
Row lengths are 2n-1 = A005408(n-1).
Removing every other column appears to give A013580.
Cf. A006134, A007318, A008289, A079309, A325347, A327481, A359907, A360005.

z = 60; p[n_, x_] := p[x] = (x^n - 1)/(x - 1); Table[p[n, x], {n, 1, z/4}]; f1[n_, x_] := f1[n, x] = Numerator[Factor[p[n, x] /. x -> x + 1/x]]; Table[Expand[f1[n, x]], {n, 0, z/4}]
Flatten[Table[CoefficientList[f1[n, x], x], {n, 1, z/4}]]

A231147_row(n) = {Vecrev(Vec(numerator((-1+(x+(1/x))^n)/(x+(1/x)-1))))} \\ John Tyler Rascoe, Sep 10 2024

A360460 Two times the median of the unordered prime signature of n; a(1) = 1.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 6, 4, 2, 2, 3, 2, 2, 2, 8, 2, 3, 2, 3, 2, 2, 2, 4, 4, 2, 6, 3, 2, 2, 2, 10, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 5, 4, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 2, 2, 2, 3, 12, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 5, 8, 2, 2, 2, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 14 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). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

			The unordered prime signature of 2520 is {1,1,2,3}, with median 3/2, so a(2520) = 3.

The version for divisors is A063655.
For mean instead of two times median we have A088529/A088530.
Prime signature is A124010, unordered A118914.
The version for prime indices is A360005.
The version for distinct prime indices is A360457.
The version for distinct prime factors is A360458.
The version for prime factors is A360459.
Positions of even terms are A360553.
Positions of odd terms are A360554.
The version for 0-prepended differences is A360555.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices.
A325347 counts partitions w/ integer median, complement A307683.
A329976 counts partitions with median multiplicity 1.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A133464, A359907, A359908, A360454.

Table[If[n==1,1,2*Median[Last/@FactorInteger[n]]],{n,100}]

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

Gus Wiseman, Jan 20 2023

nonn

			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)

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

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.

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

\\ 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

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

A360459 Two times the median of the multiset of prime factors of n; a(1) = 2.

Original entry on oeis.org

2, 4, 6, 4, 10, 5, 14, 4, 6, 7, 22, 4, 26, 9, 8, 4, 34, 6, 38, 4, 10, 13, 46, 4, 10, 15, 6, 4, 58, 6, 62, 4, 14, 19, 12, 5, 74, 21, 16, 4, 82, 6, 86, 4, 6, 25, 94, 4, 14, 10, 20, 4, 106, 6, 16, 4, 22, 31, 118, 5, 122, 33, 6, 4, 18, 6, 134, 4, 26, 10, 142, 4, 146

Offset: 1

Gus Wiseman, Feb 14 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). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

			The prime factors of 60 are {2,2,3,5}, with median 5/2, so a(60) = 5.

The union is 2 followed by A014091, complement of A014092.
The prime factors themselves are listed by A027746, distinct A027748.
The version for divisors is A063655.
Positions of odd terms are A072978 (except 1).
For mean instead of twice median: A123528/A123529, distinct A323171/A323172.
Positions of even terms are A359913 (and 1).
The version for prime indices is A360005.
The version for distinct prime indices is A360457.
The version for distinct prime factors is A360458.
The version for prime multiplicities is A360460.
The version for 0-prepended differences is A360555.
A112798 lists prime indices, length A001222, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A027336, A078174, A316413, A359907, A359908, A360006, A360007, A360248, A360552.

Table[2*Median[Join@@ConstantArray@@@FactorInteger[n]],{n,100}]

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

Gus Wiseman, Jan 21 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).

			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

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.

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

A360458 Two times the median of the set of distinct prime factors of n; a(1) = 2.

Original entry on oeis.org

2, 4, 6, 4, 10, 5, 14, 4, 6, 7, 22, 5, 26, 9, 8, 4, 34, 5, 38, 7, 10, 13, 46, 5, 10, 15, 6, 9, 58, 6, 62, 4, 14, 19, 12, 5, 74, 21, 16, 7, 82, 6, 86, 13, 8, 25, 94, 5, 14, 7, 20, 15, 106, 5, 16, 9, 22, 31, 118, 6, 122, 33, 10, 4, 18, 6, 134, 19, 26, 10, 142, 5

Offset: 1

Gus Wiseman, Feb 14 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). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

			The prime factors of 336 are {2,2,2,2,3,7}, with distinct parts {2,3,7}, with median 3, so a(336) = 6.

The union is 2 followed by A014091, complement of A014092.
Distinct prime factors are listed by A027748.
The version for divisors is A063655.
Positions of odd terms are A100367.
For mean instead of two times median we have A323171/A323172.
The version for prime indices is A360005.
The version for distinct prime indices is A360457.
The version for prime factors is A360459.
The version for prime multiplicities is A360460.
Positions of even terms are A360552.
The version for 0-prepended differences is A360555.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A078174, A316413, A325347, A359907, A360006, A360248, A360453, A360550, A360551.

Table[2*Median[First/@FactorInteger[n]],{n,100}]

A360551 Numbers > 1 whose distinct prime indices have non-integer median.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 33, 35, 36, 38, 45, 48, 51, 52, 54, 56, 58, 65, 69, 72, 74, 75, 76, 77, 86, 93, 95, 96, 98, 99, 104, 106, 108, 112, 116, 119, 122, 123, 135, 141, 142, 143, 144, 145, 148, 152, 153, 158, 161, 162, 172, 175, 177, 178, 185, 192, 196

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A325700 in having 330 and lacking 462.
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. Distinct prime indices are listed by A304038.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is not in the sequence.
The prime indices of 462 are {1,2,4,5}, with distinct parts {1,2,4,5}, with median 3, so 462 is not in the sequence.

For mean instead of median we have the complement of A326621.
Positions of odd terms in A360457.
The complement (without 1) is A360550, counted by A360686.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551 complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices, length A001221, sum A066328.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A316413, A348551, A360006, A360009, A360248, A360453.

Select[Range[2,100],!IntegerQ[Median[PrimePi/@First/@FactorInteger[#]]]&]

cp's OEIS Frontend

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

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A360245 Number of integer partitions of n where the parts have the same median as the distinct parts.

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A360254 Number of integer partitions of n with more adjacent equal parts than distinct parts.

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A231147 Array of coefficients of numerator polynomials of the rational function p(n, x + 1/x), where p(n,x) = (x^n - 1)/(x - 1).

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A360460 Two times the median of the unordered prime signature of n; a(1) = 1.

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

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A360459 Two times the median of the multiset of prime factors of n; a(1) = 2.

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

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