Displaying 1-10 of 12 results found.
Two times the median of the set of distinct prime indices of n; a(1) = 1.
+0
40
1, 2, 4, 2, 6, 3, 8, 2, 4, 4, 10, 3, 12, 5, 5, 2, 14, 3, 16, 4, 6, 6, 18, 3, 6, 7, 4, 5, 20, 4, 22, 2, 7, 8, 7, 3, 24, 9, 8, 4, 26, 4, 28, 6, 5, 10, 30, 3, 8, 4, 9, 7, 32, 3, 8, 5, 10, 11, 34, 4, 36, 12, 6, 2, 9, 4, 38, 8, 11, 6, 40, 3, 42, 13, 5, 9, 9, 4, 44, 4
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). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.
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. Distinct prime indices are listed by A304038.
EXAMPLE
The prime indices of 65 are {3,6}, with distinct parts {3,6}, with median 9/2, so a(65) = 9.
The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so a(900) = 4.
MATHEMATICA
Table[If[n==1, 1, 2*Median[PrimePi/@First/@FactorInteger[n]]], {n, 100}]
CROSSREFS
The version for divisors is A063655. The version for all prime indices is A360005. The version for distinct prime factors is A360458. The version for all prime factors is A360459. The version for prime multiplicities is A360460. Positions of even terms are A360550. Positions of odd terms are A360551. The version for 0-prepended differences is A360555. A304038 lists distinct prime indices.
Numbers for which the prime indices do not have the same median as the distinct prime indices.
+0
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 > 1 whose distinct prime indices have integer median.
+0
18
2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 42, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70, 71, 73, 78, 79, 80, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 94, 97, 100
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. Distinct prime indices are listed by A304038. 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 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is in the sequence.
The prime indices of 330 are {1,2,3,5}, with distinct parts {1,2,3,5}, with median 5/2, so 330 is not in the sequence.
MATHEMATICA
Select[Range[2, 100], IntegerQ[Median[PrimePi/@First/@FactorInteger[#]]]&]
CROSSREFS
For mean instead of median we have A326621. Positions of even terms in A360457. The complement (without 1) is A360551. Partitions with these Heinz numbers are counted by A360686.
Two times the median of the set of distinct prime factors of n; a(1) = 2.
+0
13
2, 4, 6, 4, 10, 5, 14, 4, 6, 7, 22, 5, 26, 9, 8, 4, 34, 5, 38, 7, 10, 13, 46, 5, 10, 15, 6, 9, 58, 6, 62, 4, 14, 19, 12, 5, 74, 21, 16, 7, 82, 6, 86, 13, 8, 25, 94, 5, 14, 7, 20, 15, 106, 5, 16, 9, 22, 31, 118, 6, 122, 33, 10, 4, 18, 6, 134, 19, 26, 10, 142, 5
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). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.
EXAMPLE
The prime factors of 336 are {2,2,2,2,3,7}, with distinct parts {2,3,7}, with median 3, so a(336) = 6.
MATHEMATICA
Table[2*Median[First/@FactorInteger[n]], {n, 100}]
CROSSREFS
Distinct prime factors are listed by A027748. The version for divisors is A063655. Positions of odd terms are A100367. The version for prime indices is A360005. The version for distinct prime indices is A360457. The version for prime factors is A360459. The version for prime multiplicities is A360460. Positions of even terms are A360552. The version for 0-prepended differences is A360555. A304038 lists distinct prime indices.
Cf. A000975, A026424, A078174, A316413, A325347, A359907, A360006, A360248, A360453, A360550, A360551.
Numbers > 1 whose distinct prime indices have non-integer median.
+0
13
6, 12, 14, 15, 18, 24, 26, 28, 33, 35, 36, 38, 45, 48, 51, 52, 54, 56, 58, 65, 69, 72, 74, 75, 76, 77, 86, 93, 95, 96, 98, 99, 104, 106, 108, 112, 116, 119, 122, 123, 135, 141, 142, 143, 144, 145, 148, 152, 153, 158, 161, 162, 172, 175, 177, 178, 185, 192, 196
COMMENTS
First differs from A325700 in having 330 and lacking 462. 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. Distinct prime indices are listed by A304038. 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 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is not in the sequence.
The prime indices of 462 are {1,2,4,5}, with distinct parts {1,2,4,5}, with median 3, so 462 is not in the sequence.
MATHEMATICA
Select[Range[2, 100], !IntegerQ[Median[PrimePi/@First/@FactorInteger[#]]]&]
CROSSREFS
For mean instead of median we have the complement of A326621.
Numbers > 1 whose unordered prime signature has non-integer median.
+0
13
12, 18, 20, 28, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 192, 200, 207, 208, 212, 236, 242, 244, 245, 261, 268, 272, 275, 279, 284, 288, 292, 304, 316, 320, 325, 332, 333
COMMENTS
First differs from A187039 in having 2520 and lacking 1 and 12600. A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization. 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 unordered prime signature of 2520 is {3,2,1,1}, with median 3/2, so 2520 is in the sequence.
The unordered prime signature of 12600 is {3,2,2,1}, with median 2, so 12600 is not in the sequence.
MATHEMATICA
Select[Range[2, 100], !IntegerQ[Median[Last/@FactorInteger[#]]]&]
CROSSREFS
For mean instead of median we have A070011.
Numbers > 1 whose unordered prime signature has integer median.
+0
11
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83
COMMENTS
First differs from A067340 in having 60. A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization. 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 unordered prime signature of 60 is {1,1,2}, with median 1, so 60 is in the sequence.
The unordered prime signature of 1260 is {1,1,2,2}, with median 3/2, so 1260 is not in the sequence.
MATHEMATICA
Select[Range[2, 100], IntegerQ[Median[Last/@FactorInteger[#]]]&]
CROSSREFS
Positions of even terms in A360460. The complement is A360554 (without 1). These partitions are counted by A360687. A360454 = numbers whose prime indices and signature have the same median.
Numbers for which the prime indices have the same median as the distinct prime indices.
+0
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.
+0
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 distinct parts have the same median as the multiplicities.
+0
7
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
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) = 8 partitions:
1 . . 22 221 3111 . 3311 333 3331 32222
211 41111 32211 33211 33221
42211 44111
322111 52211
511111 322211
332111
422111
3221111
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[Union[#]]&]], {n, 0, 30}]
CROSSREFS
These partitions have ranks A360453. A116608 counts partitions by number of distinct parts.
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