A359908 -id:A359908 - cp's OEIS frontend

A362619 One and all numbers whose greatest prime factor is a mode, meaning it appears at least as many times as each of the others.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, May 09 2023

nonn

First differs from A304678 in having 300.

			The prime factorization of 300 is 2*2*3*5*5, with modes {2,5} and maximum 5, so 300 is in the sequence.

Wikipedia, Mode (statistics).

Partitions of this type are counted by A171979.
The case of a unique mode is A362616, counted by A362612.
The complement is A362620, counted by A240302.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A362605 ranks partitions with a more than one mode, counted by A362607.
A362606 ranks partitions with a more than one co-mode, counted by A362609.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
A362621 ranks partitions with median equal to maximum, counted by A053263.
Cf. A000040, A002865, A237824, A237984, A327473, A327476, A359908.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],MemberQ[Commonest[prifacs[#]],Max[prifacs[#]]]&]

A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

90, 270, 525, 550, 756, 810, 1666, 1911, 1950, 2268, 2430, 2625, 2695, 2700, 2750, 5566, 6762, 6804, 6897, 7128, 7290, 8100, 8500, 9310, 9750, 10285, 10478, 11011, 11550, 11662, 12250, 12375, 12495, 13125, 13377, 13750, 14014, 14703, 18865, 19435, 20412, 21384

Offset: 1

Gus Wiseman, Jun 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.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
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 6897 are {2,5,5,8}, with mean 5, median 5, and modes {5}, so 6897 is in the sequence.
The terms together with their prime indices begin:
     90: {1,2,2,3}
    270: {1,2,2,2,3}
    525: {2,3,3,4}
    550: {1,3,3,5}
    756: {1,1,2,2,2,4}
    810: {1,2,2,2,2,3}
   1666: {1,4,4,7}
   1911: {2,4,4,6}
   1950: {1,2,3,3,6}
   2268: {1,1,2,2,2,2,4}
   2430: {1,2,2,2,2,2,3}

For just primes instead of prime powers we have A363722.
Including prime-powers gives A363727, counted by A363719.
These partitions are counted by A363728.
For unequal instead of equal we have A363730, counted by A363720.
A000961 lists the prime powers, complement A024619.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A215366, A327473, A327476, A359893, A359908, A360009, A360248, A360550, A363721, A363725, A363741.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[1000],!PrimePowerQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

A359904 Numbers whose prime factors and prime signature have the same mean.

Original entry on oeis.org

1, 4, 27, 400, 3125, 9072, 10800, 14580, 24057, 35721, 50625, 73984, 117760, 134400, 158976, 181440, 191488, 389376, 452709, 544000, 583680, 664848, 731136, 774400, 823543, 878592, 965888

Offset: 1

Gus Wiseman, Jan 25 2023

nonn

The multiset of prime factors of n is row n of A027746.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The terms together with their prime factors begin:
      1: {}
      4: {2,2}
     27: {3,3,3}
    400: {2,2,2,2,5,5}
   3125: {5,5,5,5,5}
   9072: {2,2,2,2,3,3,3,3,7}
  10800: {2,2,2,2,3,3,3,5,5}
  14580: {2,2,3,3,3,3,3,3,5}
  24057: {3,3,3,3,3,3,3,11}
  35721: {3,3,3,3,3,3,7,7}
  50625: {3,3,3,3,5,5,5,5}
  73984: {2,2,2,2,2,2,2,2,17,17}

The prime factors are A027746, mean A123528/A123529.
The prime signature is A124010, mean A088529/A088530.
For prime indices instead of factors we have A359903.
A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A078175 = numbers whose prime factors have integer mean, indices A316413.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A360005 gives median of prime indices (times two).
Cf. A240219, A326622, A327473, A359905, A359908, A359913, A360008, A360068.

prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Select[Range[1000],Mean[prifac[#]]==Mean[prisig[#]]&]

A360455 Number of integer partitions of n for which the distinct parts have the same median as the multiplicities.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 10 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(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

For mean instead of median: A114638, ranks A324570.
For parts instead of multiplicities: A360245, ranks A360249.
These partitions have ranks A360453.
For parts instead of distinct parts: A360456, ranks A360454.
A000041 counts integer partitions, strict A000009.
A116608 counts partitions by number of distinct parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A240219, A326619/A326620, A359894, A359895, A360068, A360243, A360244, A360248.

Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[Union[#]]&]],{n,0,30}]

A360952 Number of strict integer partitions of n with non-integer median; a(0) = 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 3, 0, 4, 1, 6, 1, 8, 4, 11, 5, 15, 10, 20, 13, 27, 22, 36, 28, 47, 43, 63, 56, 82, 79, 107, 103, 140, 141, 180, 181, 232, 242, 299, 308, 380, 402, 483, 511, 613, 656, 772, 824, 969, 1047, 1215, 1309, 1514, 1642, 1882, 2039, 2334, 2539, 2882

Offset: 0

Gus Wiseman, Mar 10 2023

nonn

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).

			The a(0) = 1 through a(15) = 11 partitions (0 = {}, A..E = 10..14):
  0  .  .  21  .  32  .  43  .  54  4321  65    6321  76    5432  87
                  41     52     63        74          85    6431  96
                         61     72        83          94    6521  A5
                                81        92          A3    8321  B4
                                          A1          B2          C3
                                          5321        C1          D2
                                                      5431        E1
                                                      7321        6432
                                                                  7431
                                                                  7521
                                                                  9321

The non-strict version is A307683, ranks A359912.
The non-strict complement is A325347, ranks A359908.
The strict complement is counted by A359907.
For mean instead of median we have A361391, non-strict A349156.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A067538 = partitions with integer mean, complement A102627, ranks A316413.
A359893/A359901/A359902 count partitions by median.
A360005(n)/2 ranks the median statistic.
Cf. A000975, A051293, A082550, A240219, A240850, A327475, A359897.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!IntegerQ[Median[#]]&]],{n,0,30}]

a(n) = A000009(n) - A359907(n).

A362620 Numbers whose greatest prime factor is not a mode, meaning it appears fewer times than some other.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 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

Offset: 1

Gus Wiseman, May 11 2023

nonn

First differs from A112769 in lacking 300.

			The prime factorization of 90 is 2*3*3*5, with modes {3} and maximum 5, so 90 is in the sequence.

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

Partitions of this type are counted by A240302.
The complement is A362619, counted by A171979.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A362605 ranks partitions with a more than one mode, counted by A362607.
A362606 ranks partitions with a more than one co-mode, counted by A362609.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
A362621 ranks partitions with median equal to maximum, counted by A053263.
Cf. A000040, A002865, A237824, A237984, A327473, A327476, A359908, A362616.

filter:= proc(n) local F;
  F:= sort(ifactors(n)[2], (a,b) -> a[1]Robert Israel, Dec 15 2023

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2,100],FreeQ[Commonest[prifacs[#]],Max[prifacs[#]]]&]

A362617 Numbers whose prime factorization has both (1) even length, and (2) unequal middle parts.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159, 161, 166, 177

Offset: 1

Gus Wiseman, May 10 2023

nonn

Also numbers n whose median prime factor is not a prime factor of n, where 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 factorization of 60 is 2*2*3*5, with middle parts (2,3), so 60 is in the sequence.

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

Partitions of this type are counted by A238479.
The complement (without 1) is A362618, counted by A238478.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A359893 counts partitions by median.
A359908 ranks partitions with integer median, counted by A325347.
A359912 ranks partitions with non-integer median, counted by A307683.
A362605 ranks partitions with more than one mode, counted by A362607.
A362611 counts modes in prime factorization, triangle version A362614.
A362621 ranks partitions with median equal to maximum, counted by A053263.
A362622 ranks partitions whose maximum is a middle part, counted by A237824.
Contains A006881 and (except for 1) A030229.
Cf. A000040, A171979, A327473, A327476, A356862, A359907, A362616, A362620.

filter:= proc(n) local F,m;
  F:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));
  m:= nops(F);
  m::even and F[m/2] <> F[m/2+1]
end proc:
select(filter, [$2..200]); # Robert Israel, Dec 15 2023

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2,100],FreeQ[prifacs[#],Median[prifacs[#]]]&]

A360456 Number of integer partitions of n for which the parts have the same median as the multiplicities.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 10 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(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

For mean instead of median: A360068, ranks A359903.
For distinct parts instead of multiplicities: A360245, ranks A360249.
These partitions have ranks A360454.
For distinct parts instead of parts: A360455, ranks A360453.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A114638, A240219, A359894, A359895, A360244, A360248.

Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[#]&]],{n,0,30}]

A362618 Numbers whose prime factorization has either (1) odd length, or (2) equal middle parts.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 88, 89, 90, 92, 96, 97, 98, 99, 101

Offset: 1

Gus Wiseman, May 10 2023

nonn

Also numbers n whose median prime factor is a prime factor of n.

			The prime factorization of 90 is 2*3*3*5, with middle parts (3,3), so 90 is in the sequence.

Partitions of this type are counted by A238478.
The complement (without 1) is A362617, counted by A238479.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A359893 counts partitions by median.
A359908 ranks partitions with integer median, counted by A325347.
A359912 ranks partitions with non-integer median, counted by A307683.
A362611 ranks modes in prime factorization, counted by A362614.
A362621 ranks partitions with median equal to maximum, counted by A053263.
A362622 ranks partitions whose maximum is a middle part, counted by A237824.
Cf. A000040, A171979, A327473, A327476, A356862, A359907, A360686, A360952, A362616, A362620.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2,100],MemberQ[prifacs[#],Median[prifacs[#]]]&]

A363722 Nonprime numbers whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 90, 121, 125, 128, 169, 243, 256, 270, 289, 343, 361, 512, 525, 529, 550, 625, 729, 756, 810, 841, 961, 1024, 1331, 1369, 1666, 1681, 1849, 1911, 1950, 2048, 2187, 2197, 2209, 2268, 2401, 2430, 2625, 2695, 2700, 2750, 2809

Offset: 1

Gus Wiseman, Jun 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.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
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:
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    16: {1,1,1,1}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    49: {4,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
    90: {1,2,2,3}
   121: {5,5}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}

These partitions are counted by A363719 - 1 for n > 0.
Including primes gives A363727, counted by A363719.
For prime powers instead of just primes we have A363729, counted by A363728.
For unequal instead of equal we have A363730, counted by A363720.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with multiple modes, counted by A362610.
A360005 gives twice the median of prime indices.
A362611 counts modes in prime indices, triangle A362614.
A362613 counts co-modes in prime indices, triangle A362615.
A363486 gives least mode in prime indices, A363487 greatest.
Just two statistics:
- (mean) = (median): A359889, counted by A240219.
- (mean) != (median): A359890, counted by A359894.
- (mean) = (mode): counted by A363723, see A363724, A363731.
- (median) = (mode): counted by A363740.
Cf. A000961, A327473, A327476, A359893, A359908, A360009, A360550, A363721, A363725, A363741.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Select[Range[100],!PrimeQ[#]&&{Mean[prix[#]]}=={Median[prix[#]]}==modes[prix[#]]&]

Complement of A000040 in A363727.

Assuming there is a unique mode, we have A326567(a(n))/A326568(a(n)) = A360005(a(n))/2 = A363486(a(n)) = A363487(a(n)).

cp's OEIS Frontend

A362619 One and all numbers whose greatest prime factor is a mode, meaning it appears at least as many times as each of the others.

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A363729 Numbers that are not a power of a prime but whose prime indices satisfy (mean) = (median) = (mode), assuming there is a unique mode.

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A359904 Numbers whose prime factors and prime signature have the same mean.

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A360455 Number of integer partitions of n for which the distinct parts have the same median as the multiplicities.

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A360952 Number of strict integer partitions of n with non-integer median; a(0) = 1.

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Formula

A362620 Numbers whose greatest prime factor is not a mode, meaning it appears fewer times than some other.

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Maple

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A362617 Numbers whose prime factorization has both (1) even length, and (2) unequal middle parts.

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Maple

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A360456 Number of integer partitions of n for which the parts have the same median as the multiplicities.

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Mathematica

A362618 Numbers whose prime factorization has either (1) odd length, or (2) equal middle parts.