A360457 -id:A360457 - cp's OEIS frontend

A360553 Numbers > 1 whose unordered prime signature has integer median.

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

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

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).

			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.

For mean instead of median we have A067340, complement A070011.
Positions of even terms in A360460.
The complement is A360554 (without 1).
These partitions are counted by A360687.
- 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.
A112798 lists prime indices, length A001222, sum A056239.
A124010 lists prime signature.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360454 = numbers whose prime indices and signature have the same median.
Cf. A000975, A026424, A078174, A078175, A316413, A326621, A360453.

Select[Range[2,100],IntegerQ[Median[Last/@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}]
```

A360552 Numbers > 1 whose distinct prime factors have integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103

Offset: 1

Gus Wiseman, Feb 16 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 prime factors of 900 are {2,2,3,3,5,5}, with distinct parts {2,3,5}, with median 3, so 900 is in the sequence.

For mean instead of median we have A078174, complement of A176587.
The complement is A100367 (without 1).
Positions of even terms in A360458.
- 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.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A323171/A323172 = mean of distinct prime factors, indices A326619/A326620.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A056239, A078175, A175352, A316413, A326621, A360009.

Select[Range[2,100],IntegerQ[Median[First/@FactorInteger[#]]]&]

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}]
```

A361857 Number of integer partitions of n such that the maximum is greater than twice the median.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 7, 11, 16, 25, 37, 52, 74, 101, 138, 185, 248, 325, 428, 554, 713, 914, 1167, 1476, 1865, 2336, 2922, 3633, 4508, 5562, 6854, 8405, 10284, 12536, 15253, 18489, 22376, 26994, 32507, 39038, 46802, 55963, 66817, 79582, 94643, 112315

Offset: 1

Gus Wiseman, Apr 02 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 a(5) = 1 through a(10) = 16 partitions:
  (311)  (411)   (511)    (521)     (522)      (622)
         (3111)  (4111)   (611)     (621)      (721)
                 (31111)  (4211)    (711)      (811)
                          (5111)    (5211)     (5221)
                          (32111)   (6111)     (5311)
                          (41111)   (33111)    (6211)
                          (311111)  (42111)    (7111)
                                    (51111)    (43111)
                                    (321111)   (52111)
                                    (411111)   (61111)
                                    (3111111)  (331111)
                                               (421111)
                                               (511111)
                                               (3211111)
                                               (4111111)
                                               (31111111)
The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 > 2*2, so y is counted under a(10).

For length instead of median we have A237751.
For minimum instead of median we have A237820.
The complement is counted by A361848.
The equal version is A361849, ranks A361856.
Reversing the inequality gives A361858.
Allowing equality gives A361859, ranks A361868.
These partitions have ranks A361867.
For mean instead of median we have A361907.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A013580, A027193, A061395, A237755, A237824, A240219, A361394, A361851, A361860.

Table[Length[Select[IntegerPartitions[n],Max@@#>2*Median[#]&]],{n,30}]

A361864 Number of set partitions of {1..n} whose block-medians have integer median.

Original entry on oeis.org

1, 0, 3, 6, 30, 96, 461, 2000, 10727, 57092, 342348

Offset: 1

Gus Wiseman, Apr 04 2023

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 a(1) = 1 through a(4) = 6 set partitions:
  {{1}}  .  {{123}}      {{1}{234}}
            {{13}{2}}    {{123}{4}}
            {{1}{2}{3}}  {{1}{2}{34}}
                         {{12}{3}{4}}
                         {{1}{24}{3}}
                         {{13}{2}{4}}
The set partition {{1,2},{3},{4}} has block-medians {3/2,3,4}, with median 3, so is counted under a(4).

For mean instead of median we have A361865.
For sum instead of outer median we have A361911, means A361866.
A000110 counts set partitions.
A000975 counts subsets with integer median, mean A327475.
A013580 appears to count subsets by median, A327481 by mean.
A308037 counts set partitions with integer average block-size.
A325347 counts partitions w/ integer median, complement A307683.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A027193, A067659, A079309, A231147, A359893, A359907, A361801.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],IntegerQ[Median[Median/@#]]&]],{n,6}]

A361204 Positive integers k such that 2*omega(k) <= bigomega(k).

Original entry on oeis.org

1, 4, 8, 9, 16, 24, 25, 27, 32, 36, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 100, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304

Offset: 1

Gus Wiseman, Mar 14 2023

nonn

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    16: {1,1,1,1}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    48: {1,1,1,1,2}
    49: {4,4}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}

Robert Israel, Table of n, a(n) for n = 1..10000

These partitions are counted by A237363.
The complement is A361393.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
              less: A360254, ranks A360558
             equal: A239959, ranks A067801
           greater: A237365, ranks A361393
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324517, A324521, A324522, A324560, A324562.

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(t[2],t=F) >= 2*nops(F)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 22 2023

Select[Range[100],2*PrimeNu[#]<=PrimeOmega[#]&]

A001222(a(n)) >= 2*A001221(a(n)).

A361395 Positive integers k such that 2*omega(k) >= bigomega(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74

Offset: 1

Gus Wiseman, Mar 16 2023

nonn

Differs from A068938 in having 1 and 4 and lacking 80.

Includes all squarefree numbers.

			The prime indices of 80 are {1,1,1,1,3}, with 5 parts and 2 distinct parts, and 2*2 < 5, so 80 is not in the sequence.

Complement of A360558.
Positions of nonnegative terms in A361205.
These partitions are counted by A361394.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
              less: A360254, ranks A360558
             equal: A239959, ranks A067801
           greater: A237365, ranks A361393
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324515, A324517, A324521, A324560, A324562.
Cf. A360248, A360249, A360454.

Select[Range[100],2*PrimeNu[#]>=PrimeOmega[#]&]

A001222(a(n)) <= 2*A001221(a(n)).

A361867 Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

Original entry on oeis.org

20, 28, 40, 44, 52, 56, 66, 68, 76, 78, 80, 84, 88, 92, 99, 102, 104, 112, 114, 116, 117, 120, 124, 132, 136, 138, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 190, 198, 200, 204, 207, 208, 212, 220, 222, 224, 228, 230, 232, 234

Offset: 1

Gus Wiseman, Apr 05 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 prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 > 2*(3/2), so 84 is in the sequence.
The terms together with their prime indices begin:
   20: {1,1,3}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   52: {1,1,6}
   56: {1,1,1,4}
   66: {1,2,5}
   68: {1,1,7}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   92: {1,1,9}
   99: {2,2,5}

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal version is A361856, counted by A361849.
These partitions are counted by A361857, reverse A361858.
Including the equal case gives A361868, counted by A361859.
For mean instead of median we have A361907.
A000975 counts subsets with integer median.
A001222 counts prime factors, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A111907, A237751, A237820, A359893, A361848, A361855, A361908, A361909.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], Max@@prix[#]>2*Median[prix[#]]&]

A361868 Positive integers > 1 whose prime indices satisfy (maximum) >= 2*(median).

Original entry on oeis.org

12, 20, 24, 28, 40, 42, 44, 48, 52, 56, 60, 63, 66, 68, 72, 76, 78, 80, 84, 88, 92, 96, 99, 102, 104, 112, 114, 116, 117, 120, 124, 126, 130, 132, 136, 138, 140, 144, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 189, 190, 192, 195

Offset: 1

Gus Wiseman, Apr 05 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 prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 >= 2*(3/2), so 84 is in the sequence.
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}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal case is A361856, counted by A361849.
These partitions are counted by A361859.
The unequal case is A361867, counted by A361857.
The complement is counted by A361858.
A000975 counts subsets with integer median.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
Cf. A027193, A053263, A111907, A118096, A237753, A237821, A361848, A361855, A361906, A361908.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@prix[#]>=2*Median[prix[#]]&]

cp's OEIS Frontend

A360553 Numbers > 1 whose unordered prime signature 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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Mathematica

A360552 Numbers > 1 whose distinct prime factors have integer median.

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Mathematica

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

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Mathematica

A361857 Number of integer partitions of n such that the maximum is greater than twice the median.

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Mathematica

A361864 Number of set partitions of {1..n} whose block-medians have integer median.

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Mathematica

A361204 Positive integers k such that 2*omega(k) <= bigomega(k).

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A361395 Positive integers k such that 2*omega(k) >= bigomega(k).

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A361867 Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

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