A360453
Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.
Original entry on oeis.org
1, 2, 9, 12, 18, 40, 100, 112, 125, 180, 250, 252, 300, 352, 360, 392, 396, 405, 450, 468, 504, 540, 588, 600, 612, 675, 684, 720, 756, 792, 828, 832, 882, 900, 936, 1008, 1044, 1116, 1125, 1176, 1188, 1200, 1224, 1332, 1350, 1368, 1372, 1404, 1440, 1452, 1476
Offset: 1
The terms together with their prime indices begin:
1: {}
2: {1}
9: {2,2}
12: {1,1,2}
18: {1,2,2}
40: {1,1,1,3}
100: {1,1,3,3}
112: {1,1,1,1,4}
125: {3,3,3}
180: {1,1,2,2,3}
250: {1,3,3,3}
252: {1,1,2,2,4}
300: {1,1,2,3,3}
352: {1,1,1,1,1,5}
360: {1,1,1,2,2,3}
For example, the prime indices of 756 are {1,1,2,2,2,4} with distinct parts {1,2,4} with median 2 and multiplicities {1,2,3} with median 2, so 756 is in the sequence.
For indices instead of multiplicities we have
A360249, counted by
A360245.
For indices instead of distinct indices we have
A360454, counted by
A360456.
These partitions are counted by
A360455.
A316413 = numbers whose prime indices have integer mean, distinct
A326621.
A360005 gives median of prime indices (times two).
A340830
Number of strict integer partitions of n such that every part is a multiple of the number of parts.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 1, 4, 1, 6, 1, 5, 2, 6, 1, 8, 1, 7, 4, 7, 1, 12, 1, 8, 6, 9, 1, 16, 1, 10, 9, 11, 1, 21, 1, 12, 13, 12, 1, 28, 1, 13, 17, 16, 1, 33, 1, 19, 22, 15, 1, 45, 1, 16, 28, 25, 1, 47, 1, 28, 34, 18
Offset: 1
The a(n) partitions for n = 1, 6, 10, 14, 18, 20, 24, 26, 30:
1 6 10 14 18 20 24 26 30
4,2 6,4 8,6 10,8 12,8 16,8 18,8 22,8
8,2 10,4 12,6 14,6 18,6 20,6 24,6
12,2 14,4 16,4 20,4 22,4 26,4
16,2 18,2 22,2 24,2 28,2
9,6,3 14,10 14,12 16,14
12,9,3 16,10 18,12
15,6,3 20,10
15,9,6
18,9,3
21,6,3
15,12,3
Note: A-numbers of Heinz-number sequences are in parentheses below.
The case where length divides sum also is
A340827.
The version for factorizations is
A340851.
Factorization of this type are counted by
A340853.
A072233 counts partitions by sum and length, with strict case
A008289.
A102627 counts strict partitions whose length divides sum.
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340828 counts strict partitions with length divisible by maximum.
A340829 counts strict partitions with Heinz number divisible by sum.
A276428
Sum over all partitions of n of the number of distinct parts i of multiplicity i.
Original entry on oeis.org
0, 1, 0, 1, 2, 3, 3, 6, 7, 12, 15, 22, 27, 40, 49, 68, 87, 116, 145, 193, 239, 311, 387, 494, 611, 776, 952, 1193, 1464, 1817, 2214, 2733, 3315, 4060, 4911, 5974, 7195, 8713, 10448, 12585, 15048, 18039, 21486, 25660, 30462, 36231, 42888, 50820, 59972, 70843, 83354
Offset: 0
a(5) = 3 because in the partitions [1,1,1,1,1], [1,1,1,2], [1',2',2], [1,1,3], [2,3], [1',4], [5] of 5 only the marked parts satisfy the requirement.
-
g := (sum(x^(i^2)*(1-x^i), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<1, 0, add((p-> p+`if`(i<>j, 0,
[0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60); # Alois P. Heinz, Sep 19 2016
-
b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[If[i==j, x, 1]*b[n - i*j, i-1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; a[n_] := (row = T[n]; row.Range[0, Length[row]-1]); Table[a[n], {n, 0, 60}] // Flatten (* Jean-François Alcover, Nov 28 2016, after Alois P. Heinz's Maple code for A276427 *)
-
apply( A276428(n,s,c)={forpart(p=n,c=1;for(i=1,#p,p[i]==if(i<#p, p[i+1])&&c++&&next; c==p[i]&&s++; c=1));s}, [0..20]) \\ M. F. Hasler, Oct 27 2019
A360454
Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.
Original entry on oeis.org
1, 2, 9, 54, 100, 120, 125, 135, 168, 180, 189, 240, 252, 264, 280, 297, 300, 312, 336, 351, 396, 408, 440, 450, 456, 459, 468, 480, 513, 520, 528, 540, 552, 560, 588, 612, 616, 621, 624, 672, 680, 684, 696, 728, 744, 756, 760, 783, 816, 828, 837, 880, 882
Offset: 1
The terms together with their prime indices begin:
1: {}
2: {1}
9: {2,2}
54: {1,2,2,2}
100: {1,1,3,3}
120: {1,1,1,2,3}
125: {3,3,3}
135: {2,2,2,3}
168: {1,1,1,2,4}
180: {1,1,2,2,3}
189: {2,2,2,4}
240: {1,1,1,1,2,3}
For example, the prime indices of 336 are {1,1,1,1,2,4} with median 1 and multiplicities {1,1,4} with median 1, so 336 is in the sequence.
For distinct indices instead of indices we have
A360453, counted by
A360455.
For distinct indices instead of multiplicities:
A360249, counted by
A360245.
These partitions are counted by
A360456.
A240219 counts partitions with mean equal to median, ranked by
A359889.
A359894 counts partitions with mean different from median, ranks
A359890.
A360005 gives median of prime indices (times two).
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Median[prix[#]]==Median[Length/@Split[prix[#]]]&]
A360455
Number of integer partitions of n for which the distinct parts have the same median as the multiplicities.
Original entry on oeis.org
1, 1, 0, 0, 2, 1, 1, 0, 2, 2, 5, 8, 10, 14, 20, 19, 26, 31, 35, 41, 55, 65, 85, 102, 118, 151, 181, 201, 236, 281, 313, 365, 424, 495, 593, 688, 825, 978, 1181, 1374, 1650, 1948, 2323, 2682, 3175, 3680, 4314, 4930, 5718, 6546, 7532, 8557, 9777, 11067, 12622
Offset: 0
The a(1) = 1 through a(11) = 8 partitions:
1 . . 22 221 3111 . 3311 333 3331 32222
211 41111 32211 33211 33221
42211 44111
322111 52211
511111 322211
332111
422111
3221111
These partitions have ranks
A360453.
A116608 counts partitions by number of distinct parts.
-
Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[Union[#]]&]],{n,0,30}]
A360456
Number of integer partitions of n for which the parts have the same median as the multiplicities.
Original entry on oeis.org
1, 1, 0, 0, 1, 0, 0, 1, 2, 5, 7, 10, 14, 21, 28, 36, 51, 64, 84, 106, 132, 165, 202, 252, 311, 391, 473, 579, 713, 868, 1069, 1303, 1617, 1954, 2404, 2908, 3556, 4282, 5200, 6207, 7505, 8934, 10700, 12717, 15165, 17863, 21222, 24976, 29443, 34523, 40582, 47415
Offset: 0
The a(1) = 1 through a(11) = 10 partitions:
1 . . 22 . . 2221 3311 333 4222 5222
32111 3222 33211 33221
32211 42211 52211
42111 43111 53111
321111 52111 62111
421111 322211
3211111 431111
521111
4211111
32111111
These partitions have ranks
A360454.
A008284 counts partitions by number of parts.
-
Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[#]&]],{n,0,30}]
A360682
Number of integer partitions of n of length > 2 whose second differences have median 0.
Original entry on oeis.org
0, 0, 0, 1, 1, 1, 5, 4, 10, 13, 18, 23, 44, 44, 72, 98, 132, 162, 241, 277, 394, 497, 643, 800, 1076, 1287, 1660, 2078, 2604, 3192, 4065, 4892, 6113, 7490, 9166, 11110, 13717, 16429, 20033, 24201, 29143, 34945, 42251, 50219, 60253, 71852, 85503, 101501, 120899
Offset: 0
The a(3) = 1 through a(9) = 13 partitions:
(111) (1111) (11111) (222) (22111) (2222) (333)
(321) (31111) (3221) (432)
(2211) (211111) (3311) (531)
(21111) (1111111) (22211) (22221)
(111111) (32111) (33111)
(41111) (51111)
(221111) (222111)
(311111) (321111)
(2111111) (411111)
(11111111) (2211111)
(3111111)
(21111111)
(111111111)
For first differences we have
A237363.
For sum instead of median we have
A360683.
A008284 counts partitions by number of parts.
A360005 gives median of prime indices (times two).
-
Table[Length[Select[IntegerPartitions[n],Median[Differences[#,2]]==0&]],{n,0,30}]
A243149
Number of compositions of n such that the sum of the parts counted without multiplicities is equal to the sum of all multiplicities.
Original entry on oeis.org
1, 1, 0, 0, 4, 3, 4, 0, 11, 31, 70, 177, 242, 382, 482, 874, 1655, 4440, 10696, 24390, 49867, 95850, 172980, 289229, 492233, 811753, 1468084, 2813206, 5929361, 12780690, 27858421, 59275097, 122326098, 243179349, 467856049, 873044584, 1588187110, 2842593612
Offset: 0
a(8) = 11: [1,1,3,3], [1,3,1,3], [1,3,3,1], [3,1,1,3], [3,1,3,1], [3,3,1,1], [1,1,1,1,4], [1,1,1,4,1], [1,1,4,1,1], [1,4,1,1,1], [4,1,1,1,1].
Cf.
A114638 (the same for partitions).
-
b:= proc(n, i, p) option remember; `if`(n=0, p!,
`if`(i<1, 0, expand(add(x^`if`(j=0, 0, i-j)*
b(n-i*j, i-1, p+j)/j!, j=0..n/i))))
end:
a:= n-> coeff(b(n$2, 0), x, 0):
seq(a(n), n=0..50);
-
b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, Expand[Sum[x^If[j == 0, 0, i - j]*b[n - i*j, i - 1, p + j]/j!, {j,0, n/i}]]]];
a[n_] := Coefficient[b[n, n, 0], x, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 21 2018, translated from Maple *)
A360683
Number of integer partitions of n whose second differences sum to 0, meaning either there is only one part, or the first two parts have the same difference as the last two parts.
Original entry on oeis.org
1, 1, 2, 3, 4, 4, 8, 6, 11, 12, 17, 14, 32, 23, 40, 44, 64, 59, 104, 93, 149, 157, 218, 227, 342, 349, 481, 538, 713, 777, 1052, 1145, 1494, 1692, 2130, 2416, 3064, 3449, 4286, 4918, 6028, 6882, 8424, 9620, 11634, 13396, 16022, 18416, 22019, 25248, 29954
Offset: 0
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) (22111) (71)
(321) (1111111) (2222)
(2211) (3221)
(111111) (3311)
(22211)
(221111)
(11111111)
For mean instead of sum we have a(n) -
A008619(n).
For median instead of sum we have
A360682.
A008284 counts partitions by number of parts.
-
Table[Length[Select[IntegerPartitions[n],Total[Differences[#,2]]==0&]],{n,0,30}]
A363220
Number of integer partitions of n whose conjugate has the same median.
Original entry on oeis.org
1, 0, 1, 1, 1, 3, 3, 8, 8, 12, 12, 15, 21, 27, 36, 49, 65, 85, 112, 149, 176, 214, 257, 311, 378, 470, 572, 710, 877, 1080, 1322, 1637, 1983, 2416, 2899, 3465, 4107, 4891, 5763, 6820, 8071, 9542, 11289, 13381, 15808, 18710, 22122, 26105, 30737, 36156, 42377
Offset: 1
The partition y = (4,3,1,1) has median 2, and its conjugate (4,2,2,1) also has median 2, so y is counted under a(9).
The a(1) = 1 through a(9) = 8 partitions:
(1) . (21) (22) (311) (321) (511) (332) (333)
(411) (4111) (422) (711)
(3111) (31111) (611) (4221)
(3311) (4311)
(4211) (6111)
(5111) (51111)
(41111) (411111)
(311111) (3111111)
For mean instead of median we have
A047993.
Median of conjugate by rank is
A363219.
These partitions are ranked by
A363261.
A122111 represents partition conjugation.
A325347 counts partitions with integer median.
A352491 gives n minus Heinz number of conjugate.
Cf.
A000975,
A067538,
A114638,
A360068,
A360242,
A360248,
A362617,
A362618,
A362621,
A363223,
A363260.
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Median[#]==Median[conj[#]]&]],{n,30}]
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