A360006 -id:A360006 - cp's OEIS frontend

A359913 Numbers whose multiset of prime factors has integer median.

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Jan 25 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The terms together with their prime factors begin:
   2: {2}
   3: {3}
   4: {2,2}
   5: {5}
   7: {7}
   8: {2,2,2}
   9: {3,3}
  11: {11}
  12: {2,2,3}
  13: {13}
  15: {3,5}
  16: {2,2,2,2}
  17: {17}
  18: {2,3,3}
  19: {19}
  20: {2,2,5}
  21: {3,7}
  23: {23}
  24: {2,2,2,3}

Prime factors are listed by A027746.
The complement is A072978, for prime indices A359912.
For mean instead of median we have A078175, for prime indices A316413.
For prime indices instead of factors we have A359908, counted by A325347.
Positions of even terms in A360005.
A067340 lists numbers whose prime signature has integer mean.
A112798 lists prime indices, length A001222, sum A056239.
A325347 counts partitions with integer median, strict A359907.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A067538, A175352, A348551, A359890, A359905, A360007, A360006, A360069.

Select[Range[2,100],IntegerQ[Median[Flatten[ConstantArray@@@FactorInteger[#]]]]&]

A360616 Half the number of prime factors of n (counted with multiplicity, A001222), rounded down.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 2, 0, 1, 1, 3, 1, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 2, 2, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 08 2023

nonn

			The prime indices of 378 are {1,2,2,2,4}, so a(378) = floor(5/2) = 2.

Positions of 0's are 1 and A000040.
Positions of first appearances are A000302 = 2^(2k) for k >= 0.
Positions of 1's are A168645.
Rounding up instead of down gives A360617.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A001248, A026424, A280076, A359912, A360006, A360457, A360674.

Mathematica
```
Table[Floor[PrimeOmega[n]/2],{n,100}]
```

A360008 Positions of first appearances in the sequence giving the mean of prime indices (A326567/A326568).

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 12, 13, 14, 17, 18, 19, 23, 24, 26, 29, 31, 37, 38, 41, 42, 43, 47, 48, 52, 53, 54, 58, 59, 61, 67, 71, 72, 73, 74, 76, 79, 83, 86, 89, 92, 96, 97, 101, 103, 104, 106, 107, 108, 109, 113, 122, 124, 127, 131, 137, 139, 142, 148, 149, 151, 152

Offset: 1

Gus Wiseman, Jan 24 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:
    1: {}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   11: {5}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   17: {7}
   18: {1,2,2}
   19: {8}
   23: {9}
   24: {1,1,1,2}

Positions of first appearances in A326567/A326568.
The version for median instead of mean is A360007, unsorted A360006.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A326567/A326568 gives mean of prime indices.
A359908 = numbers w/ integer median of prime indices, complement A359912.
Cf. A026424, A051293, A327473, A348551, A359889, A360005.

nn=1000;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[If[n==1,1,Mean[prix[n]]],{n,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

A360617 Half the number of prime factors of n (counted with multiplicity, A001222), rounded up.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 08 2023

nonn

			The prime indices of 378 are {1,2,2,2,4}, so a(378) = ceiling(5/2) = 3.

Positions of 0's and 1's are 1 and A037143.
Positions of first appearances are A081294.
Rounding down instead of up gives A360616.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000040, A000302, A001248, A026424, A168645, A359912, A360006, A360457.

Mathematica
```
Table[Ceiling[PrimeOmega[n]/2],{n,100}]
```

A359892 Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not have the same mean as median.

Original entry on oeis.org

12, 18, 20, 28, 42, 44, 45, 48, 50, 52, 63, 66, 68, 70, 72, 75, 76, 78, 80, 92, 98, 99, 102, 108, 112, 114, 116, 117, 120, 124, 130, 138, 147, 148, 153, 154, 162, 164, 165, 168, 170, 171, 172, 174, 175, 176, 180, 182, 186, 188, 190, 192, 195, 200, 207, 208

Offset: 1

Gus Wiseman, Jan 22 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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The terms together with their prime indices begin:
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   28: {1,1,4}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   70: {1,3,4}
   72: {1,1,1,2,2}
For example, the prime indices of 180 are {1,1,2,2,3}, with mean 9/5 and median 2, so 180 is in the sequence.

A subset of A026424 = numbers with odd bigomega.
The LHS (mean of prime indices) is A326567/A326568.
This is the odd-length case of A359890, complement A359889.
The complement is A359891.
These partitions are counted by A359896, complement A359895.
The RHS (median of prime indices) is A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A359902 counts odd-length partitions by median.
Cf. A240219, A327473, A327476, A348551, A359894, A359898, A359899, A359900, A359911, A359912, A360006-A360009.

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

Intersection of A026424 and A359890.

A360953 Numbers whose right half of prime indices (exclusive) adds up to half the total sum of prime indices.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 30, 48, 49, 63, 64, 70, 81, 108, 121, 154, 165, 169, 192, 256, 270, 273, 286, 289, 325, 361, 442, 529, 561, 567, 595, 625, 646, 675, 729, 741, 750, 768, 841, 874, 931, 961, 972, 1024, 1045, 1173, 1334, 1369, 1495, 1575, 1653, 1681, 1750

Offset: 1

Gus Wiseman, Mar 09 2023

nonn

Also numbers whose left half of prime indices (inclusive) adds up to half the total sum of prime indices.

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:
     1: {}
     4: {1,1}
     9: {2,2}
    12: {1,1,2}
    16: {1,1,1,1}
    25: {3,3}
    30: {1,2,3}
    48: {1,1,1,1,2}
    49: {4,4}
    63: {2,2,4}
    64: {1,1,1,1,1,1}
    70: {1,3,4}
    81: {2,2,2,2}
   108: {1,1,2,2,2}
For example, the prime indices of 1575 are {2,2,3,3,4}, with right half (exclusive) {3,4}, with sum 7, and the total sum of prime indices is 14, so 1575 is in the sequence.

The left version is A056798.
The inclusive version is A056798.
These partitions are counted by A360674.
The left inclusive version is A360953 (this sequence).
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000005, A000040, A001248, A026424, A359912, A360006, A360616, A360617, A360671, A360673.

Select[Range[100],With[{w=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[Take[w,-Floor[Length[w]/2]]]==Total[w]/2]&]

A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 6, 30, 42, 49, 60, 66, 70, 78, 84, 90, 102, 105, 114, 120, 126, 132, 138, 140, 150, 154, 156, 168, 174, 186, 198, 204, 210, 222, 228, 234, 246, 258, 264, 270, 276, 280, 282, 286, 294, 306, 308, 312, 315, 318, 330, 342, 348, 350, 354, 366, 372, 378, 385

Offset: 1

Gus Wiseman, Feb 19 2023

nonn

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).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
   30: {1,2,3}
   42: {1,2,4}
   49: {4,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with median 1. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with median 1/2. So 2760 is not in the sequence.

For distinct prime indices instead of 0-prepended differences: A360453.
For mean instead of median we have A360680.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Multisets with integer median:
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
Cf. A000975, A026424, A316413, A340610, A359904, A360006, A360248, A360558, A360687, A360688.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Median[Length/@Split[prix[#]]] == Median[Differences[Prepend[prix[#],0]]]&]

cp's OEIS Frontend

A359913 Numbers whose multiset of prime factors has integer median.

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Mathematica

A360616 Half the number of prime factors of n (counted with multiplicity, A001222), rounded down.

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A360008 Positions of first appearances in the sequence giving the mean of prime indices (A326567/A326568).

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A360617 Half the number of prime factors of n (counted with multiplicity, A001222), rounded up.

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A359892 Members of A026424 (numbers with an odd number of prime factors) whose prime indices do not have the same mean as median.

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Formula

A360953 Numbers whose right half of prime indices (exclusive) adds up to half the total sum of prime indices.

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Mathematica

A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

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