A362619 -id:A362619 - cp's OEIS frontend

A237824 Number of partitions of n such that 2*(least part) >= greatest part.

1, 2, 3, 4, 5, 7, 7, 10, 11, 13, 14, 19, 18, 23, 25, 29, 30, 38, 37, 46, 48, 54, 57, 70, 69, 80, 85, 97, 100, 118, 118, 137, 144, 159, 168, 193, 195, 220, 233, 259, 268, 303, 311, 348, 367, 399, 419, 469, 483, 532, 560, 610, 639, 704, 732, 801, 841, 908, 954

Offset: 1

Clark Kimberling, Feb 16 2014

nonn

By conjugation, also the number of integer partitions of n whose greatest part appears at a middle position, namely at k/2, (k+1)/2, or (k+2)/2 where k is the number of parts. These partitions have ranks A362622. - Gus Wiseman, May 14 2023

			a(6) = 7 counts these partitions:  6, 42, 33, 222, 2211, 21111, 111111.
From _Gus Wiseman_, May 14 2023: (Start)
The a(1) = 1 through a(8) = 10 partitions such that 2*(least part) >= greatest part:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (42)      (322)      (53)
                    (1111)  (2111)   (222)     (2221)     (332)
                            (11111)  (2211)    (22111)    (422)
                                     (21111)   (211111)   (2222)
                                     (111111)  (1111111)  (22211)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part appears at a middle position:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (2211)    (2221)     (332)
                                     (111111)  (1111111)  (2222)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..300 from John Tyler Rascoe)

Cf. A237821, A118096, A053263.
The complement is counted by A237820, ranks A362982.
For modes instead of middles we have A362619, counted by A171979.
These partitions have ranks A362981.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
Cf. A002865, A008284, A237984, A238478, A238479, A327472, A359893, A362612, A362622, A384426.

z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] == Max[p]], {n, z}] (* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* this sequence *)
(* or *)
nmax = 100; Rest[CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j,k,2*k}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 13 2025 *)
(* or *)
nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(2*k - 1))*(1 - x^(2*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax+1)]; s += x^k/(1 - x^k)/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 14 2025 *)

N=60; x='x+O('x^N);
gf = sum(m=1, N, (x^m)/(1-x^m)) + sum(i=1, N, sum(j=1, i, x^((2*i)+j)/prod(k=0, j, 1 - x^(k+i))));
Vec(gf) \\ John Tyler Rascoe, Mar 07 2024

G.f.: Sum_{m>0} x^m/(1-x^m) + Sum_{i>0} Sum_{j=1..i} x^((2*i)+j) / Product_{k=0..j} (1 - x^(k+i)). - John Tyler Rascoe, Mar 07 2024
G.f.: Sum_{k>=1} x^k / Product_{j=k..2*k} (1 - x^j). - Vaclav Kotesovec, Jun 13 2025
a(n) ~ phi^(3/2) * exp(Pi*sqrt(2*n/15)) / (5^(1/4) * sqrt(2*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 14 2025

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

Original entry on oeis.org

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}]]]&]

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[#]]]&]

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])&]

A362981 Heinz numbers of integer partitions such that 2*(least part) >= greatest part.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 55, 59, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 91, 96, 97, 101, 103, 105, 107, 108, 109, 113, 119, 121, 125

Offset: 1

Gus Wiseman, May 14 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

By conjugation, also Heinz numbers of partitions whose greatest part appears at a middle position, namely k/2, (k+1)/2, or (k+2)/2, where k is the number of parts. These partitions have ranks A362622.

			The terms together with their prime indices begin:
     1: {}         16: {1,1,1,1}      36: {1,1,2,2}
     2: {1}        17: {7}            37: {12}
     3: {2}        18: {1,2,2}        41: {13}
     4: {1,1}      19: {8}            43: {14}
     5: {3}        21: {2,4}          45: {2,2,3}
     6: {1,2}      23: {9}            47: {15}
     7: {4}        24: {1,1,1,2}      48: {1,1,1,1,2}
     8: {1,1,1}    25: {3,3}          49: {4,4}
     9: {2,2}      27: {2,2,2}        53: {16}
    11: {5}        29: {10}           54: {1,2,2,2}
    12: {1,1,2}    31: {11}           55: {3,5}
    13: {6}        32: {1,1,1,1,1}    59: {17}
    15: {2,3}      35: {3,4}          61: {18}

For prime factors instead of indices we have A081306.
Prime indices are listed by A112798, length A001222, sum A056239.
The complement is A362982, counted by A237820.
Partitions of this type are counted by A237824.
Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362616, A362619, A362621, A362622.

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

A362982 Heinz numbers of partitions such that 2*(least part) < greatest part.

Original entry on oeis.org

10, 14, 20, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 56, 57, 58, 60, 62, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 115, 116, 117, 118, 120, 122, 123, 124, 126

Offset: 1

Gus Wiseman, May 14 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    10: {1,3}        44: {1,1,5}      70: {1,3,4}
    14: {1,4}        46: {1,9}        74: {1,12}
    20: {1,1,3}      50: {1,3,3}      76: {1,1,8}
    22: {1,5}        51: {2,7}        78: {1,2,6}
    26: {1,6}        52: {1,1,6}      80: {1,1,1,1,3}
    28: {1,1,4}      56: {1,1,1,4}    82: {1,13}
    30: {1,2,3}      57: {2,8}        84: {1,1,2,4}
    33: {2,5}        58: {1,10}       85: {3,7}
    34: {1,7}        60: {1,1,2,3}    86: {1,14}
    38: {1,8}        62: {1,11}       87: {2,10}
    39: {2,6}        66: {1,2,5}      88: {1,1,1,5}
    40: {1,1,1,3}    68: {1,1,7}      90: {1,2,2,3}
    42: {1,2,4}      69: {2,9}        92: {1,1,9}

For prime factors instead of indices we have A069900, complement A081306.
Prime indices are listed by A112798, length A001222, sum A056239.
Partitions of this type are counted by A237820.
The complement is A362981, counted by A237824.
Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362616, A362619, A362621, A362622.

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

A363223 Numbers with bigomega equal to median prime index.

Original entry on oeis.org

2, 9, 10, 50, 70, 75, 105, 110, 125, 130, 165, 170, 175, 190, 195, 230, 255, 275, 285, 290, 310, 325, 345, 370, 410, 425, 430, 435, 465, 470, 475, 530, 555, 575, 590, 610, 615, 645, 670, 686, 705, 710, 725, 730, 775, 790, 795, 830, 885, 890, 915, 925, 970

Offset: 1

Gus Wiseman, May 29 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).

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:
    2: {1}
    9: {2,2}
   10: {1,3}
   50: {1,3,3}
   70: {1,3,4}
   75: {2,3,3}
  105: {2,3,4}
  110: {1,3,5}
  125: {3,3,3}
  130: {1,3,6}
  165: {2,3,5}
  170: {1,3,7}
  175: {3,3,4}

For maximum instead of median we have A106529, counted by A047993.
For minimum instead of median we have A324522, counted by A006141.
Partitions of this type are counted by A361800.
For twice median we have A362050, counted by A362049.
For maximum instead of length we have A362621, counted by A053263.
A000975 counts subsets with integer median.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A359908 lists numbers whose prime indices have integer median.
A360005 gives twice median of prime indices.
Cf. A000040, A013580, A079309, A240219, A327473, A327476, A361860, A362619, A362622, A362980.

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

2*A001222(a(n)) = A360005(a(n)).

cp's OEIS Frontend

A237824 Number of partitions of n such that 2*(least part) >= greatest part.

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

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

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A362981 Heinz numbers of integer partitions such that 2*(least part) >= greatest part.

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A362982 Heinz numbers of partitions such that 2*(least part) < greatest part.

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A363223 Numbers with bigomega equal to median prime index.

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