A053263 -id:A053263 - cp's OEIS frontend

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

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[#]]]&]

A363130 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k non-co-modes, all 0's removed.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 4, 3, 8, 3, 6, 9, 10, 12, 11, 18, 1, 15, 24, 3, 13, 37, 6, 25, 43, 9, 19, 64, 18, 29, 81, 25, 33, 99, 44, 42, 129, 59, 1, 39, 162, 93, 3, 62, 201, 116, 6, 55, 247, 175, 13, 81, 303, 224, 19, 84, 364, 309, 35, 103, 457, 389, 53, 105, 535, 529, 86

Offset: 0

Gus Wiseman, May 18 2023

We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.

			Triangle begins:
   1
   1
   2
   3
   4   1
   4   3
   8   3
   6   9
  10  12
  11  18   1
  15  24   3
  13  37   6
  25  43   9
  19  64  18
  29  81  25
  33  99  44
Row n = 9 counts the following partitions:
  (9)          (441)       (32211)
  (54)         (522)
  (63)         (711)
  (72)         (3222)
  (81)         (3321)
  (333)        (4221)
  (432)        (4311)
  (531)        (5211)
  (621)        (6111)
  (222111)     (22221)
  (111111111)  (33111)
               (42111)
               (51111)
               (321111)
               (411111)
               (2211111)
               (3111111)
               (21111111)

Row sums are A000041.
Row lengths are approximately A000196.
Column k = 0 is A047966.
For modes instead of non-co-modes we have A362614, rank stat A362611.
For co-modes instead of non-co-modes we have A362615, rank stat A362613.
For non-modes instead of non-co-modes we have A363126, rank stat A363127.
Columns k > 1 sum to A363128.
Column k = 1 is A363129.
This rank statistic (number of non-co-modes) is A363131.
A008284/A058398 count partitions by length/mean.
A275870 counts collapsible partitions.
A353836 counts partitions by number of distinct run-sums.
A359893 counts partitions by median.
Cf. A002865, A053263, A098859, A237984, A327472, A353863, A362612, A363124, A363125.

ncomsi[ms_]:=Select[Union[ms],Count[ms,#]>Min@@Length/@Split[ms]&];
DeleteCases[Table[Length[Select[IntegerPartitions[n] , Length[ncomsi[#]]==k&]],{n,0,15},{k,0,Sqrt[n]}],0,{2}]

A361861 Number of integer partitions of n where the median is twice the minimum.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 5, 5, 8, 11, 16, 20, 28, 38, 53, 67, 87, 111, 146, 183, 236, 297, 379, 471, 591, 729, 909, 1116, 1376, 1682, 2065, 2507, 3055, 3699, 4482, 5395, 6501, 7790, 9345, 11153, 13316, 15839, 18844, 22333, 26466, 31266, 36924, 43478, 51177

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(4) = 1 through a(11) = 11 partitions:
  (31)  (221)  (321)  (421)   (62)     (621)    (442)     (542)
                      (2221)  (521)    (4221)   (721)     (821)
                              (3221)   (4311)   (5221)    (6221)
                              (3311)   (22221)  (5311)    (6311)
                              (22211)  (32211)  (32221)   (33221)
                                                (33211)   (42221)
                                                (42211)   (43211)
                                                (222211)  (52211)
                                                          (222221)
                                                          (322211)
                                                          (2222111)
The partition (3,2,2,2,1,1) has median 2 and minimum 1, so is counted under a(11).
The partition (5,4,2) has median 4 and minimum 2, so is counted under a(11).

For maximum instead of median we have A118096.
For length instead of median we have A237757, without the coefficient A006141.
With minimum instead of twice minimum we have A361860.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A027193, A039900, A053263, A067659, A111907, A116608, A237753, A237755, A237824, A361848, A361853.

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

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])&]

A361800 Number of integer partitions of n with the same length as median.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 1, 2, 3, 3, 3, 3, 4, 6, 9, 13, 14, 15, 18, 21, 27, 32, 40, 46, 55, 62, 72, 82, 95, 111, 131, 157, 186, 225, 264, 316, 366, 430, 495, 578, 663, 768, 880, 1011, 1151, 1316, 1489, 1690, 1910, 2158, 2432, 2751, 3100, 3505, 3964, 4486, 5079, 5764

Offset: 1

Gus Wiseman, Apr 07 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(15) = 9 partitions (A=10, B=11):
  1  .  .  22  .  .  331  332  333  433  533  633  733   833   933
           31             431  432  532  632  732  832   932   A32
                               531  631  731  831  931   A31   B31
                                                   4441  4442  4443
                                                         5441  5442
                                                         5531  5532
                                                               6441
                                                               6531
                                                               6621

For minimum instead of median we have A006141, for twice minimum A237757.
For maximum instead of median we have A047993, for twice length A237753.
For maximum instead of length we have A053263, for twice median A361849.
For mean instead of median we have A206240 (zeros removed).
For minimum instead of length we have A361860.
For twice median we have A362049, ranks A362050.
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.
Cf. A008284, A013580, A027193, A079309, A240219, A362048.

Table[Length[Select[IntegerPartitions[n],Length[#]==Median[#]&]],{n,30}]

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[#]

A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 15, 14, 32, 34, 65, 55, 150, 100, 225, 237, 425, 296, 824, 489, 1267, 1133, 1809, 1254, 4018, 2142, 4499, 4550, 7939, 4564, 14571, 6841, 18285, 16047, 23408, 17495, 52545, 21636, 49943, 51182, 92516, 44582, 144872, 63260, 175318, 169232, 205353

Offset: 0

Gus Wiseman, May 23 2023

nonn

Equivalently, n = (length)*(minimum).

			The a(2) = 1 through a(7) = 14 partitions:
  (31)  (321)  (62)    (32221)  (93)      (3222221)
        (411)  (3221)  (33211)  (552)     (3322211)
               (3311)  (42211)  (642)     (3332111)
               (4211)  (43111)  (732)     (4222211)
               (5111)  (52111)  (822)     (4322111)
                       (61111)  (322221)  (4331111)
                                (332211)  (4421111)
                                (333111)  (5222111)
                                (422211)  (5321111)
                                (432111)  (5411111)
                                (441111)  (6221111)
                                (522111)  (6311111)
                                (531111)  (7211111)
                                (621111)  (8111111)
                                (711111)

Removing the factor 2 gives A099777.
Taking maximum instead of mean and including odd indices gives A118096.
For length instead of mean and including odd indices we have A237757.
For (maximum) = 2*(mean) see A361851, A361852, A361853, A361854, A361855.
For median instead of mean we have A361861.
These partitions have ranks A363133.
For maximum instead of minimum we have A363218.
For median instead of minimum we have A363224.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
A268192 counts partitions by complement size, ranks A326844.
Cf. A053263, A111907, A237753, A237755, A237824, A327482, A349156, A361906, A361907, A363134.

Table[Length[Select[IntegerPartitions[2n],2*Min@@#==Mean[#]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A363132(n): return sum(1 for s,p in partitions(n<<1,m=n,size=True) if n==s*min(p,default=0)) if n else 0 # Chai Wah Wu, Sep 21 2023

a(31)-a(46) from Chai Wah Wu, Sep 21 2023

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

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

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A363130 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k non-co-modes, all 0's removed.

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A361861 Number of integer partitions of n where the median is twice the minimum.

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

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A361800 Number of integer partitions of n with the same length as median.

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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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A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

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A363223 Numbers with bigomega equal to median prime index.

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