A362620 -id:A362620 - cp's OEIS frontend

A362621 One and numbers whose multiset of prime factors (with multiplicity) has the same median as maximum.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 151, 157, 162, 163, 167, 169

Offset: 1

Gus Wiseman, May 12 2023

nonn

First differs from A334965 in having 750 and lacking 2250.

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 108 is 2*2*3*3*3, and the multiset {2,2,3,3,3} has median 3 and maximum 3, so 108 is in the sequence.
The prime factorization of 2250 is 2*3*3*5*5*5, and the multiset {2,3,3,5,5,5} has median 4 and maximum 5, so 2250 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           25: {3,3}           64: {1,1,1,1,1,1}
     2: {1}          27: {2,2,2}         67: {19}
     3: {2}          29: {10}            71: {20}
     4: {1,1}        31: {11}            73: {21}
     5: {3}          32: {1,1,1,1,1}     75: {2,3,3}
     7: {4}          37: {12}            79: {22}
     8: {1,1,1}      41: {13}            81: {2,2,2,2}
     9: {2,2}        43: {14}            83: {23}
    11: {5}          47: {15}            89: {24}
    13: {6}          49: {4,4}           97: {25}
    16: {1,1,1,1}    50: {1,3,3}         98: {1,4,4}
    17: {7}          53: {16}           101: {26}
    18: {1,2,2}      54: {1,2,2,2}      103: {27}
    19: {8}          59: {17}           107: {28}
    23: {9}          61: {18}           108: {1,1,2,2,2}

Partitions of this type are counted by A053263.
For mode instead of median we have A362619, counted by A171979.
For parts at middle position (instead of median) we have A362622.
The complement is A362980, counted by A237821.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
Cf. A000040, A002865, A237824, A327473, A327476, A359908, A362616, A362620.

Select[Range[100],(y=Flatten[Apply[ConstantArray,FactorInteger[#],{1}]];Max@@y==Median[y])&]

A362622 One and numbers whose prime factorization has its greatest part at a middle position.

Original entry on oeis.org

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, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91

Offset: 1

Gus Wiseman, May 12 2023

nonn

			The prime factorization of 150 is 5*5*3*2, with middle parts {3,5}, so 150 is in the sequence.
The prime factorization of 90 is 5*3*3*2, with middle parts {3,3}, so 90 is not in the sequence.

Partitions of this type are counted by A237824.
For modes instead of middles we have A362619, counted by A171979.
The version for median instead of middles is A362621, counted by A053263.
The complement for median is A362980, counted by A237821.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A362611 counts modes in prime factorization.
A362613 counts co-modes in prime factorization.
Cf. A000040, A002865, A237984, A327473, A327476, A359908, A362616, A362620.

mpm[q_]:=MemberQ[If[OddQ[Length[q]],{Median[q]},{q[[Length[q]/2]],q[[Length[q]/2+1]]}],Max@@q];
Select[Range[100],#==1||mpm[Flatten[Apply[ConstantArray,FactorInteger[#],{1}]]]&]

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.

Original entry on oeis.org

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[#]]]&]

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[#]]]&]

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[#]]]&]

A362980 Numbers whose multiset of prime factors (with multiplicity) has different median from maximum.

Original entry on oeis.org

6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 104, 105, 106, 110

Offset: 1

Gus Wiseman, May 12 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 factorization of 108 is 2*2*3*3*3, and the multiset {2,2,3,3,3} has median 3 and maximum 3, so 108 is not in the sequence.
The prime factorization of 2250 is 2*3*3*5*5*5, and the multiset {2,3,3,5,5,5} has median 4 and maximum 5, so 2250 is in the sequence.
The terms together with their prime indices begin:
     6: {1,2}        36: {1,1,2,2}      60: {1,1,2,3}
    10: {1,3}        38: {1,8}          62: {1,11}
    12: {1,1,2}      39: {2,6}          63: {2,2,4}
    14: {1,4}        40: {1,1,1,3}      65: {3,6}
    15: {2,3}        42: {1,2,4}        66: {1,2,5}
    20: {1,1,3}      44: {1,1,5}        68: {1,1,7}
    21: {2,4}        45: {2,2,3}        69: {2,9}
    22: {1,5}        46: {1,9}          70: {1,3,4}
    24: {1,1,1,2}    48: {1,1,1,1,2}    72: {1,1,1,2,2}
    26: {1,6}        51: {2,7}          74: {1,12}
    28: {1,1,4}      52: {1,1,6}        76: {1,1,8}
    30: {1,2,3}      55: {3,5}          77: {4,5}
    33: {2,5}        56: {1,1,1,4}      78: {1,2,6}
    34: {1,7}        57: {2,8}          80: {1,1,1,1,3}
    35: {3,4}        58: {1,10}         82: {1,13}

Partitions of this type are counted by A237821.
For mode instead of median we have A362620, counted by A240302.
The complement is A362621, counted by A053263.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
Cf. A000040, A002865, A171979, A237824, A237984, A327473, A327476, A359908, A362616, A362619, A362622.

Select[Range[100],(y=Flatten[Apply[ConstantArray,FactorInteger[#],{1}]];Max@@y!=Median[y])&]

cp's OEIS Frontend

A362621 One and numbers whose multiset of prime factors (with multiplicity) has the same median as maximum.

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A362622 One and numbers whose prime factorization has its greatest part at a middle position.

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

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

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A362980 Numbers whose multiset of prime factors (with multiplicity) has different median from maximum.

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