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Numbers for which the prime indices do not have the same median as the distinct prime indices.
+10
24
12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 200
COMMENTS
First differs from A242416 in lacking 180, with prime indices {1,1,2,2,3}. First differs from A360246 in lacking 126 and having 1950. 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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The terms together with their prime indices begin:
12: {1,1,2}
18: {1,2,2}
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}
50: {1,3,3}
52: {1,1,6}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
63: {2,2,4}
68: {1,1,7}
72: {1,1,1,2,2}
The prime indices of 126 are {1,2,2,4} with median 2 and distinct prime indices {1,2,4} with median 2, so 126 is not in the sequence.
The prime indices of 1950 are {1,2,3,3,6} with median 3 and distinct prime indices {1,2,3,6} with median 5/2, so 1950 is in the sequence.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Median[prix[#]]!=Median[Union[prix[#]]]&]
CROSSREFS
These partitions are counted by A360244. For multiplicities instead of parts: complement of A360453. For multiplicities instead of distinct parts: complement of A360454. The complement for mean instead of median is A360247, counted by A360243. A360005 gives median of prime indices (times two).
Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.
+10
12
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
COMMENTS
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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
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.
MATHEMATICA
Select[Range[100], #==1||Median[Last/@FactorInteger[#]]== Median[PrimePi/@First/@FactorInteger[#]]&]
CROSSREFS
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).
Number of integer partitions of n with integer median of 0-appended first differences.
+10
11
1, 1, 3, 4, 5, 7, 12, 18, 25, 32, 46, 62, 79, 109, 142, 189, 240, 322, 405, 522, 671, 853, 1053, 1345, 1653, 2081, 2551, 3174, 3878, 4826, 5851, 7219, 8747, 10712, 12936, 15719, 18876, 22872, 27365, 32926, 39253, 47070, 55857, 66676, 79029, 93864, 110832
COMMENTS
Includes all partitions of odd length ( A027193). The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(1) = 1 through a(8) = 18 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(21) (22) (41) (42) (43) (44)
(111) (211) (221) (222) (61) (62)
(1111) (311) (321) (322) (332)
(11111) (411) (331) (422)
(21111) (421) (431)
(111111) (511) (521)
(3211) (611)
(22111) (2222)
(31111) (3221)
(211111) (4211)
(1111111) (22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
For example, the partition y = (3,2,2,1) has 0-appended parts (3,2,2,1,0), with differences (1,0,1,1), and the multiset {0,1,1,1} has median 1, so y is counted under a(8).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], IntegerQ[Median[Differences[Prepend[Reverse[#], 0]]]]&]], {n, 30}]
CROSSREFS
A008284 counts partitions by number of parts.
Numbers for which the prime indices have the same median as the distinct prime indices.
+10
10
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 121, 122, 123, 125, 126, 127, 128, 129, 130
COMMENTS
First differs from A072774 in having 90. First differs from A242414 in having 180. 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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The prime indices of 126 are {1,2,2,4} with median 2 and distinct prime indices {1,2,4} with median 2, so 126 is in the sequence.
The prime indices of 180 are {1,1,2,2,3} with median 2 and distinct prime indices {1,2,3} with median 2, so 180 is in the sequence.
MAPLE
isA360249 := proc(n)
local ifs, pidx, pe, medAll, medDist ;
if n = 1 then
return true ;
end if ;
ifs := ifactors(n)[2] ;
pidx := [] ;
for pe in ifs do
numtheory[pi](op(1, pe)) ;
pidx := [op(pidx), seq(%, i=1..op(2, pe))] ;
end do:
medAll := stats[describe, median](sort(pidx)) ;
pidx := convert(convert(pidx, set), list) ;
medDist := stats[describe, median](sort(pidx)) ;
if medAll = medDist then
true;
else
false;
end if;
end proc:
for n from 1 to 130 do
if isA360249(n) then
printf("%d, ", n) ;
end if;
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Median[prix[#]]==Median[Union[prix[#]]]&]
CROSSREFS
These partitions are counted by A360245. The complement for mean instead of median is A360246, counted by A360242. For multiplicities instead of distinct parts: A360454, counted by A360456. A360005 gives median of prime indices (times two).
Numbers for which the prime multiplicities (or sorted signature) have the same median as the prime indices.
+10
9
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
COMMENTS
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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
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.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Median[prix[#]]==Median[Length/@Split[prix[#]]]&]
CROSSREFS
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).
Number of integer partitions of n for which the parts have the same median as the multiplicities.
+10
5
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
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
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
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[#]&]], {n, 0, 30}]
CROSSREFS
These partitions have ranks A360454. A008284 counts partitions by number of parts.
Number of integer partitions of n with non-integer median of 0-prepended first differences.
+10
2
0, 1, 0, 1, 2, 4, 3, 4, 5, 10, 10, 15, 22, 26, 34, 42, 57, 63, 85, 105, 121, 149, 202, 230, 305, 355, 459, 544, 687, 778, 991, 1130, 1396, 1598, 1947, 2258, 2761, 3143, 3820, 4412, 5330, 6104, 7404, 8499, 10105, 11694, 13922, 15917, 18904, 21646, 25462, 29213
COMMENTS
All of these partitions have even length.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(1) = 0 through a(10) = 10 partitions:
. (11) . (31) (32) (33) (52) (53) (54) (55)
(2111) (51) (2221) (71) (72) (73)
(2211) (4111) (3311) (3222) (91)
(3111) (5111) (6111) (3322)
(321111) (3331)
(4411)
(5311)
(7111)
(322111)
(421111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Differences[Prepend[Reverse[#], 0]]]]&]], {n, 30}]
CROSSREFS
The complement is counted by A360688. A008284 counts partitions by number of parts.
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