A364056 -id:A364056 - cp's OEIS frontend

A124944 Table, number of partitions of n with k as high median.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 6, 4, 1, 1, 1, 1, 1, 8, 6, 3, 1, 1, 1, 1, 1, 11, 8, 5, 1, 1, 1, 1, 1, 1, 15, 11, 7, 3, 1, 1, 1, 1, 1, 1, 20, 15, 9, 5, 1, 1, 1, 1, 1, 1, 1, 26, 21, 12, 8, 3, 1, 1, 1, 1, 1, 1, 1, 35, 27, 16, 10, 5, 1, 1, 1, 1, 1, 1, 1, 1, 45, 37, 21, 13, 8, 3

Offset: 1

Franklin T. Adams-Watters, Nov 13 2006

For a multiset with an odd number of elements, the high median is the same as the median. For a multiset with an even number of elements, the high median is the larger of the two central elements.
This table may be read as an upper right triangle with n >= 1 as column index and k >= 1 as row index. - Peter Munn, Jul 16 2017
Arrange the parts of a partition nonincreasing order.  Remove the last part, then the first, then the last remaining part, then the first remaining part, and continue until only a single number, the high median, remains. - Clark Kimberling, May 14 2019

			For the partition [2,1^2], the sole middle element is 1, so that is the high median. For [3,2,1^2], the two middle elements are 1 and 2; the high median is the larger, 2.
From _Gus Wiseman_, Jul 12 2023: (Start)
Triangle begins:
   1
   1  1
   1  1  1
   2  1  1  1
   3  1  1  1  1
   4  3  1  1  1  1
   6  4  1  1  1  1  1
   8  6  3  1  1  1  1  1
  11  8  5  1  1  1  1  1  1
  15 11  7  3  1  1  1  1  1  1
  20 15  9  5  1  1  1  1  1  1  1
  26 21 12  8  3  1  1  1  1  1  1  1
  35 27 16 10  5  1  1  1  1  1  1  1  1
  45 37 21 13  8  3  1  1  1  1  1  1  1  1
  58 48 29 16 11  5  1  1  1  1  1  1  1  1  1
Row n = 8 counts the following partitions:
  (611)       (521)    (431)   (44)  (53)  (62)  (71)  (8)
  (5111)      (422)    (332)
  (41111)     (4211)   (3311)
  (32111)     (3221)
  (311111)    (2222)
  (221111)    (22211)
  (2111111)
  (11111111)
(End)

Row sums are A000041.
Column k = 1 is A027336(n-1), ranks A364056.
Column k = 1 in the low version is A027336, ranks A363488.
The low version of this triangle is A124943.
The rank statistic for this triangle is A363942, low version A363941.
A version for mean instead of median is A363946, low A363945.
A version for mode instead of median is A363953, low A363952.
A008284 counts partitions by length, maximum, or decreasing mean.
A026794 counts partitions by minimum, strict A026821.
A325347 counts partitions with integer median, ranks A359908.
A359893 and A359901 count partitions by median.
A360005(n)/2 returns median of prime indices.
Cf. A008289, A025065, A027193, A067538, A237984, A240219, A362608, A363740, A363943, A363944.

Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 1)/2]]] &, IntegerPartitions[#]]] &, Range[13]]  (* Peter J. C. Moses, May 14 2019 *)

A363488 Even numbers whose prime factorization has at least as many 2's as non-2's.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 84, 86, 88, 92, 94, 96, 100, 104, 106, 112, 116, 118, 120, 122, 124, 128, 132, 134, 136, 140, 142, 144, 146, 148, 152, 156, 158, 160

Offset: 1

Gus Wiseman, Jul 06 2023

nonn

The multiset of prime factors of n is row n of A027746.

Also numbers whose prime factors have low median 2, where the low median (see A124943) is either the middle part (for odd length), or the least of the two middle parts (for even length).

			The terms together with their prime indices begin:
     2: {1}            34: {1,7}             72: {1,1,1,2,2}
     4: {1,1}          36: {1,1,2,2}         74: {1,12}
     6: {1,2}          38: {1,8}             76: {1,1,8}
     8: {1,1,1}        40: {1,1,1,3}         80: {1,1,1,1,3}
    10: {1,3}          44: {1,1,5}           82: {1,13}
    12: {1,1,2}        46: {1,9}             84: {1,1,2,4}
    14: {1,4}          48: {1,1,1,1,2}       86: {1,14}
    16: {1,1,1,1}      52: {1,1,6}           88: {1,1,1,5}
    20: {1,1,3}        56: {1,1,1,4}         92: {1,1,9}
    22: {1,5}          58: {1,10}            94: {1,15}
    24: {1,1,1,2}      60: {1,1,2,3}         96: {1,1,1,1,1,2}
    26: {1,6}          62: {1,11}           100: {1,1,3,3}
    28: {1,1,4}        64: {1,1,1,1,1,1}    104: {1,1,1,6}
    32: {1,1,1,1,1}    68: {1,1,7}          106: {1,16}

Partitions of this type are counted by A027336.
The case without high median > 1 is A072978.
For mode instead of median we have A360015, high A360013.
Positions of 1's in A363941.
For mean instead of median we have A363949, high A000079.
The high version is A364056, positions of 1's in A363942.
A067538 counts partitions with integer mean, ranks A316413.
A112798 lists prime indices, length A001222, sum A056239.
A124943 counts partitions by low median, high A124944.
A363943 gives low mean of prime indices, triangle A363945.
Cf. A025065, A215366, A238495, A326567/A326568, A344296, A359889, A359908, A360005, A363486, A363487, A363944.

Select[Range[100],EvenQ[#]&&PrimeOmega[#]<=2*FactorInteger[#][[1,2]]&]

A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 14, 19, 27, 36, 51, 66, 90, 118, 156, 201, 264, 336, 434, 550, 700, 880, 1112, 1385, 1733, 2149, 2666, 3283, 4049, 4956, 6072, 7398, 9009, 10922, 13237, 15970, 19261, 23147, 27790, 33260, 39776, 47425, 56497, 67133, 79685, 94371, 111653

Offset: 1

Clark Kimberling, Feb 27 2014

Also the number of integer partitions of n + 1 with median > 1, or with no more 1's than non-1 parts. - Gus Wiseman, Jul 10 2023

			a(6) = 9 counts all the 11 partitions of 6 except 42 and 411.
From _Gus Wiseman_, Jul 10 2023 (Start)
The a(2) = 1 through a(8) = 14 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)
       (21)  (22)  (32)   (33)    (43)    (44)
             (31)  (41)   (42)    (52)    (53)
                   (221)  (51)    (61)    (62)
                          (222)   (322)   (71)
                          (321)   (331)   (332)
                          (2211)  (421)   (422)
                                  (2221)  (431)
                                  (3211)  (521)
                                          (2222)
                                          (3221)
                                          (3311)
                                          (4211)
                                          (22211)
(End)

Cf. A096373.
For mean instead of median we have A000065, ranks A057716.
The complement is counted by A027336, ranks A364056.
Rows sums of A359893 if we remove the first column.
These partitions have ranks A364058.
A000041 counts integer partitions.
A008284 counts partitions by length, A058398 by mean.
A025065 counts partitions with low mean 1, ranks A363949.
A124943 counts partitions by low median, high A124944.
A241131 counts partitions with low mode 1, ranks A360015.
Cf. A000070, A000975, A002865, A110618, A237984, A363488.

Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Length[p] + Min[p]]], {n, 50}]
Table[Length[Select[IntegerPartitions[n+1],Median[#]>1&]],{n,30}] (* Gus Wiseman, Jul 10 2023 *)

From Gus Wiseman, Jul 11 2023: (Start)
a(n>2) = A000041(n) - A096373(n-2).
a(n>1) = A000041(n-2) + A002865(n+1).
a(n) = A000041(n+1) - A027336(n).
(End)

Formula corrected by Gus Wiseman, Jul 11 2023

A364058 Heinz numbers of integer partitions with median > 1. Numbers whose multiset of prime factors has median > 2.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, Jul 14 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

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 terms together with their prime indices begin:
     3: {2}        23: {9}          42: {1,2,4}
     5: {3}        25: {3,3}        43: {14}
     6: {1,2}      26: {1,6}        45: {2,2,3}
     7: {4}        27: {2,2,2}      46: {1,9}
     9: {2,2}      29: {10}         47: {15}
    10: {1,3}      30: {1,2,3}      49: {4,4}
    11: {5}        31: {11}         50: {1,3,3}
    13: {6}        33: {2,5}        51: {2,7}
    14: {1,4}      34: {1,7}        53: {16}
    15: {2,3}      35: {3,4}        54: {1,2,2,2}
    17: {7}        36: {1,1,2,2}    55: {3,5}
    18: {1,2,2}    37: {12}         57: {2,8}
    19: {8}        38: {1,8}        58: {1,10}
    21: {2,4}      39: {2,6}        59: {17}
    22: {1,5}      41: {13}         60: {1,1,2,3}

For mean instead of median we have A057716, counted by A000065.
These partitions are counted by A238495.
The complement is A364056, counted by A027336, low version A363488.
A000975 counts subsets with integer median, A051293 for mean.
A124943 counts partitions by low median, high version A124944.
A360005 gives twice the median of prime indices, A360459 for prime factors.
A359893 and A359901 count partitions by median.
Cf. A002865, A316413, A325347, A327473, A327476, A363727.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Median[prifacs[#]]>2&]

A360005(a(n)) > 1.

A360459(a(n)) > 2.

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A124944 Table, number of partitions of n with k as high median.

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A363488 Even numbers whose prime factorization has at least as many 2's as non-2's.

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A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

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A364058 Heinz numbers of integer partitions with median > 1. Numbers whose multiset of prime factors has median > 2.

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