A360252
Numbers for which the prime indices have greater mean than the distinct prime indices.
Original entry on oeis.org
18, 50, 54, 75, 98, 108, 147, 150, 162, 242, 245, 250, 294, 324, 338, 350, 363, 375, 450, 486, 490, 500, 507, 578, 588, 605, 648, 686, 722, 726, 735, 750, 845, 847, 867, 882, 972, 1014, 1029, 1050, 1058, 1078, 1083, 1125, 1183, 1210, 1250, 1274, 1350, 1372
Offset: 1
The terms together with their prime indices begin:
18: {1,2,2}
50: {1,3,3}
54: {1,2,2,2}
75: {2,3,3}
98: {1,4,4}
108: {1,1,2,2,2}
147: {2,4,4}
150: {1,2,3,3}
162: {1,2,2,2,2}
242: {1,5,5}
245: {3,4,4}
250: {1,3,3,3}
294: {1,2,4,4}
324: {1,1,2,2,2,2}
For example, the prime indices of 350 are {1,3,3,4} with mean 11/4, and the distinct prime indices are {1,3,4} with mean 8/3, so 350 is in the sequence.
These partitions are counted by
A360250.
A316413 lists numbers whose indices have integer mean, distinct
A326621.
Cf.
A000975,
A051293,
A058398,
A067340,
A067538,
A324570,
A327482,
A359903,
A360005,
A360241,
A360248.
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prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]>Mean[Union[prix[#]]]&]
A360253
Numbers for which the prime indices have lesser mean than the distinct prime indices.
Original entry on oeis.org
12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212, 220
Offset: 1
The terms together with their prime indices begin:
12: {1,1,2}
20: {1,1,3}
24: {1,1,1,2}
28: {1,1,4}
40: {1,1,1,3}
44: {1,1,5}
45: {2,2,3}
48: {1,1,1,1,2}
52: {1,1,6}
56: {1,1,1,4}
60: {1,1,2,3}
63: {2,2,4}
68: {1,1,7}
72: {1,1,1,2,2}
For example, the prime indices of 350 are {1,3,3,4} with mean 11/4, and the distinct prime indices are {1,3,4} with mean 8/3, so 350 is not in the sequence.
These partitions are counted by
A360251.
A316413 lists numbers whose indices have integer mean, distinct
A326621.
Cf.
A000975,
A051293,
A058398,
A067340,
A067538,
A324570,
A327482,
A359903,
A360005,
A360241,
A360248.
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]
A370245
Coefficient of x^n in the expansion of ( 1/(1-x)^2 * (1+x^2)^3 )^n.
Original entry on oeis.org
1, 2, 16, 110, 828, 6352, 49696, 393668, 3148316, 25362992, 205519616, 1673272702, 13677016932, 112165564656, 922490228032, 7605558361960, 62839438825244, 520180768020464, 4313251202569216, 35818392770702104, 297846498752214128, 2479748570715505472
Offset: 0
-
a(n, s=2, t=3, u=2) = sum(k=0, n\s, binomial(t*n, k)*binomial((u+1)*n-s*k-1, n-s*k));
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.
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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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