A363943 -id:A363943 - cp's OEIS frontend

A124943 Table read by rows: number of partitions of n with k as low median.

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 2, 0, 0, 1, 6, 3, 1, 0, 0, 1, 8, 4, 2, 0, 0, 0, 1, 11, 6, 3, 1, 0, 0, 0, 1, 15, 8, 4, 2, 0, 0, 0, 0, 1, 20, 12, 5, 3, 1, 0, 0, 0, 0, 1, 26, 16, 7, 4, 2, 0, 0, 0, 0, 0, 1, 35, 22, 10, 5, 3, 1, 0, 0, 0, 0, 0, 1, 45, 29, 14, 6, 4, 2, 0, 0, 0, 0, 0, 0, 1, 58, 40, 19, 8, 5, 3, 1

Offset: 1

Franklin T. Adams-Watters, Nov 13 2006

For a multiset with an odd number of elements, the low median is the same as the median. For a multiset with an even number of elements, the low median is the smaller of the two central elements.

Arrange the parts of a partition nonincreasing order. Remove the first part, then the last, then the first remaining part, then the last remaining part, and continue until only a single number, the low median, remains. - Clark Kimberling, May 16 2019

			For the partition [2,1^2], the sole middle element is 1, so that is the low median. For [3,2,1^2], the two middle elements are 1 and 2; the low median is the smaller, 1.
First 8 rows:
  1
  1   1
  2   0   1
  3   1   0   1
  4   2   0   0   1
  6   3   1   0   0   1
  8   4   2   0   0   0   1
  11  6   3   1   0   0   0   1
From _Gus Wiseman_, Jul 09 2023: (Start)
Row n = 8 counts the following partitions:
  (71)        (62)     (53)   (44)  .  .  .  (8)
  (611)       (521)    (431)
  (5111)      (422)    (332)
  (4211)      (3221)
  (41111)     (2222)
  (3311)      (22211)
  (32111)
  (311111)
  (221111)
  (2111111)
  (11111111)
(End)

Row sums are A000041.
Column k = 1 is A027336, ranks A363488.
The high version of this triangle is A124944.
The rank statistic for this triangle is A363941, high version A363942.
A version for mean instead of median is A363945, rank statistic A363943.
A high version for mean instead of median is A363946, rank stat A363944.
A version for mode instead of median is A363952, high A363953.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A325347 counts partitions with integer median, ranks A359908.
A359893 and A359901 count partitions by median.
A360005(n)/2 returns median of prime indices.
Cf. A025065, A026794, A027193, A067538, A237984, A240219, A362608, A363740.

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

A363486 Low mode in the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 23 2023

nonn

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.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
Extending the terminology of A124943, the "low mode" in a multiset is its least mode.

Positions of first appearances are 1 and A000040.
Positions of 1's are A360013, counted by A241131.
For greatest instead of least we have A363487.
The version for median is A363941, triangle A124943.
The high version for median is A363942, triangle A124944.
The version for mean instead of mode is A363943, high A363944.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A356862 ranks partitions with a unique mode, counted by A362608.
A362605 ranks partitions with more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A362616 ranks partitions (max part) = (unique mode), counted by A362612.
Cf. A000961, A215366, A327473, A327476, A360015, A363723.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[If[n==1,0,First[modes[prix[n]]]],{n,30}]

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

Original entry on oeis.org

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 *)

A363487 High mode in the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 04 2023

nonn

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.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
Extending the terminology of A124944, the "high mode" in a multiset is its greatest mode.

Positions of first appearances are 1 and A000040.
Positions of 1's are A360015, counted by A241131.
For low instead of high mode we have A363486.
The version for low median is A363941, triangle A124943.
The version for high median is A363942, triangle A124944.
The version for mean instead of mode is A363944, low A363943.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A356862 ranks partitions with a unique mode, counted by A362608.
A362605 ranks partitions with more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A362616 ranks partitions (max part) = (unique mode), counted by A362612.
Cf. A000961, A215366, A327473, A327476, A359908, A360013, A363723, A363729.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[If[n==1,0,Last[modes[prix[n]]]],{n,30}]

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

Gus Wiseman, Jun 30 2023

nonn

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.

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

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.

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}]

A363942 High median in the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 01 2023

nonn

The high median (see A124944) in a multiset is either the middle part (for odd length), or the greatest of the two middle parts (for even length).

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.

			The prime indices of 90 are {1,2,2,3}, with high median 2, so a(90) = 2.
The prime indices of 150 are {1,2,3,3}, with high median 3, so a(150) = 3.

Positions of first appearances are 1 and A000040.
The triangle for this statistic (high median) is A124944, low A124943.
Regular median of prime indices is A360005(n)/2.
For mode instead of median we have A363487, low A363486.
The low version is A363941.
For mean instead of median we have A363944, triangle A363946, low A363943.
A061395 give maximum prime index, A055396 minimum.
A112798 lists prime indices, length A001222, sum A056239.
A362611 counts modes in prime indices, triangle A362614.
Cf. A025065, A215366, A316413, A326567/A326568, A359889, A359908, A359912, A363727, A363949, A363950.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
merr[y_]:=If[Length[y]==0,0, If[OddQ[Length[y]],y[[(Length[y]+1)/2]],y[[1+Length[y]/2]]]];
Table[merr[prix[n]],{n,100}]

A363941 Low median in the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 01 2023

nonn

The low median (see A124943) in a multiset is either the middle part (for odd length), or the least of the two middle parts (for even length).

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.

			The prime indices of 90 are {1,2,2,3}, with low median 2, so a(90) = 2.
The prime indices of 150 are {1,2,3,3}, with low median 2, so a(150) = 2.

Positions of first appearances are 1 and A000040.
The triangle for this statistic (low median) is A124943, high A124944.
Median of prime indices is A360005(n)/2.
For mode instead of median we have A363486, high A363487.
Positions of 1's are A363488.
The high version is A363942.
A067538 counts partitions with integer mean, ranked by A316413.
A112798 lists prime indices, length A001222, sum A056239.
A363943 gives low mean of prime indices, triangle A363945.
A363944 gives high mean of prime indices, triangle A363946.
Cf. A025065, A215366, A326567/A326568, A344296, A359889, A359908, A363727, A363740.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mell[y_]:=If[Length[y]==0,0, If[OddQ[Length[y]],y[[(Length[y]+1)/2]],y[[Length[y]/2]]]];
Table[mell[prix[n]],{n,30}]

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

Gus Wiseman, Jul 02 2023

nonn

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.

			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}

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.

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

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

Gus Wiseman, Jun 30 2023

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

			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)

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.

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

Gus Wiseman, Jun 30 2023

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

			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)

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.

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

cp's OEIS Frontend

A124943 Table read by rows: number of partitions of n with k as low median.

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A363486 Low mode in the multiset of prime indices of n.

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

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A363487 High mode in the multiset of prime indices of n.

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

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A363942 High median in the multiset of prime indices of n.

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A363941 Low median in the multiset of prime indices of n.

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

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

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