A326840 -id:A326840 - cp's OEIS frontend

A326836 Heinz numbers of integer partitions whose maximum part divides their sum.

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 70, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 150, 151

Offset: 1

Gus Wiseman, Jul 26 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose maximum prime index divides their sum of prime indices.

The enumeration of these partitions by sum is given by A067538.

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}

Cf. A018818, A047993, A056239, A061395, A067538, A112798, A316413.

Cf. A326837, A326838, A326839/A326840, A326841, A326843, A326850.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],Divisible[Total[primeMS[#]],Max[primeMS[#]]]&]

A359912 Numbers whose prime indices do not have integer median.

Original entry on oeis.org

1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 60, 65, 69, 74, 77, 84, 86, 93, 95, 106, 119, 122, 123, 132, 141, 142, 143, 145, 150, 156, 158, 161, 177, 178, 185, 196, 201, 202, 204, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 276, 278

Offset: 1

Gus Wiseman, Jan 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.

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:
   1: {}
   6: {1,2}
  14: {1,4}
  15: {2,3}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  51: {2,7}
  58: {1,10}
  60: {1,1,2,3}

For prime factors instead of indices we have A072978, complement A359913.
These partitions are counted by A307683.
For mean instead of median: A348551, complement A316413, counted by A349156.
The complement is A359908, counted by A325347.
Positions of odd terms in A360005.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A026424, A051293, A067538, A175352, A175761, A289509, A359890, A359905, A360006, A359907, A360009.

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

A349156 Number of integer partitions of n whose mean is not an integer.

Original entry on oeis.org

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752

Offset: 0

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

By conjugation, also the number of integer partitions of n with greatest part not dividing n.

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)

Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The complement is counted by A067538, ranked by A316413.
The geometric version is !A067539, strict !A326625, ranked by !A326623.
The strict case is !A102627.
The version for prime factors is A175352, complement A078175.
The version for distinct prime factors is A176587, complement A078174.
The ordered version (compositions) is !A271654, ranked by !A096199.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A000041 counts integer partitions.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A236634 counts unbalanced partitions, complement of A047993.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

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

a(n > 0) = A000041(n) - A067538(n).

A348551 Heinz numbers of integer partitions whose mean is not an integer.

Original entry on oeis.org

1, 6, 12, 14, 15, 18, 20, 24, 26, 33, 35, 36, 38, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 58, 60, 63, 65, 66, 69, 70, 72, 74, 75, 76, 77, 80, 86, 92, 93, 95, 96, 102, 104, 106, 108, 112, 114, 117, 119, 120, 122, 123, 124, 126, 130, 132, 135, 136, 140, 141, 142

Offset: 1

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

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 and their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  42: {1,2,4}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}

A version counting nonempty subsets is A000079 - A051293.
A version counting factorizations is A001055 - A326622.
A version counting compositions is A011782 - A271654.
A version for prime factors is A175352, complement A078175.
A version for distinct prime factors A176587, complement A078174.
The complement is A316413, counted by A067538, strict A102627.
The geometric version is the complement of A326623.
The conjugate version is the complement of A326836.
These partitions are counted by A349156.
A000041 counts partitions.
A001222 counts prime factors with multiplicity.
A018818 counts partitions into divisors, ranked by A326841.
A143773 counts partitions into multiples of the length, ranked by A316428.
A236634 counts unbalanced partitions.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement A237984.
Cf. A067539, A096199, A098743, A175397, A175761, A289508, A289509, A290103, A326028, A326645, A326837.

q:= n-> (l-> nops(l)=0 or irem(add(i, i=l), nops(l))>0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..142])[];  # Alois P. Heinz, Nov 19 2021

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!IntegerQ[Mean[primeMS[#]]]&]

A359913 Numbers whose multiset of prime factors has integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Jan 25 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 terms together with their prime factors begin:
   2: {2}
   3: {3}
   4: {2,2}
   5: {5}
   7: {7}
   8: {2,2,2}
   9: {3,3}
  11: {11}
  12: {2,2,3}
  13: {13}
  15: {3,5}
  16: {2,2,2,2}
  17: {17}
  18: {2,3,3}
  19: {19}
  20: {2,2,5}
  21: {3,7}
  23: {23}
  24: {2,2,2,3}

Prime factors are listed by A027746.
The complement is A072978, for prime indices A359912.
For mean instead of median we have A078175, for prime indices A316413.
For prime indices instead of factors we have A359908, counted by A325347.
Positions of even terms in A360005.
A067340 lists numbers whose prime signature has integer mean.
A112798 lists prime indices, length A001222, sum A056239.
A325347 counts partitions with integer median, strict A359907.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A067538, A175352, A348551, A359890, A359905, A360007, A360006, A360069.

Select[Range[2,100],IntegerQ[Median[Flatten[ConstantArray@@@FactorInteger[#]]]]&]

A326839 Numerator of A056239(n)/A061395(n) where A056239 is sum of prime indices and A061395 is maximum prime index.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 2, 1, 5, 5, 4, 1, 5, 1, 5, 3, 6, 1, 5, 2, 7, 3, 3, 1, 2, 1, 5, 7, 8, 7, 3, 1, 9, 4, 2, 1, 7, 1, 7, 7, 10, 1, 3, 2, 7, 9, 4, 1, 7, 8, 7, 5, 11, 1, 7, 1, 12, 2, 6, 3, 8, 1, 9, 11, 2, 1, 7, 1, 13, 8, 5, 9, 3, 1, 7, 4, 14, 1, 2, 10

Offset: 1

Gus Wiseman, Jul 26 2019

This is a dual form of the average of an integer partition specified by its Heinz number: A326567/A326568.

			The sequence of fractions begins: 0, 1, 1, 2, 1, 3/2, 1, 3, 2, 4/3, 1, 2, 1, 5/4, 5/3, 4, 1, 5/2, 1, 5/3.

Denominators are A326840.
Positions of 1's are A000040.
Cf. A001222, A047993, A056239, A061395, A112798, A316413.
Cf. A326567/A326568, A326836, A326845, A326848, A326850.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Numerator[Table[Total[primeMS[n]]/Max@@primeMS[n],{n,100}]]

A361391 Number of strict integer partitions of n with non-integer mean.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 4, 2, 4, 5, 11, 0, 17, 15, 13, 15, 37, 18, 53, 24, 48, 78, 103, 23, 111, 152, 143, 123, 255, 110, 339, 238, 372, 495, 377, 243, 759, 845, 873, 414, 1259, 842, 1609, 1383, 1225, 2281, 2589, 1285, 2827, 2518, 3904, 3836, 5119, 3715, 4630

Offset: 0

Gus Wiseman, Mar 11 2023

nonn

Are 1, 2, 4, 6, 12 the only zeros?

			The a(3) = 1 through a(11) = 11 partitions:
  {2,1}  .  {3,2}  .  {4,3}    {4,3,1}  {5,4}  {5,3,2}    {6,5}
            {4,1}     {5,2}    {5,2,1}  {6,3}  {5,4,1}    {7,4}
                      {6,1}             {7,2}  {6,3,1}    {8,3}
                      {4,2,1}           {8,1}  {7,2,1}    {9,2}
                                               {4,3,2,1}  {10,1}
                                                          {5,4,2}
                                                          {6,3,2}
                                                          {6,4,1}
                                                          {7,3,1}
                                                          {8,2,1}
                                                          {5,3,2,1}

The strict complement is counted by A102627.
The non-strict version is ranked by A348551, complement A316413.
The non-strict version is counted by A349156, complement A067538.
For median instead of mean we have A360952, complement A359907.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A307683 counts partitions with non-integer median, ranks A359912.
A325347 counts partitions with integer median, ranks A359908.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement of A237984.
A327475 counts subsets with integer mean.
Cf. A051293, A082550, A143773, A175397, A175761, A240219, A240850, A326027, A326641, A326849, A359897.

a:= proc(m) option remember; local b; b:=
      proc(n, i, t) option remember; `if`(i*(i+1)/2Alois P. Heinz, Mar 16 2023

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

a(31)-a(55) from Alois P. Heinz, Mar 16 2023

A363219 Twice the median of the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

0, 2, 2, 4, 2, 3, 2, 6, 4, 2, 2, 4, 2, 2, 4, 8, 2, 5, 2, 2, 3, 2, 2, 5, 4, 2, 6, 2, 2, 4, 2, 10, 2, 2, 4, 6, 2, 2, 2, 2, 2, 3, 2, 2, 6, 2, 2, 6, 4, 4, 2, 2, 2, 7, 4, 2, 2, 2, 2, 4, 2, 2, 4, 12, 3, 2, 2, 2, 2, 4, 2, 7, 2, 2, 6, 2, 4, 2, 2, 2, 8, 2, 2, 3, 2, 2

Offset: 1

Gus Wiseman, May 25 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). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

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 partition (4,2,1) has Heinz number 42 and conjugate (3,2,1,1) with median 3/2, so a(42) = 3.

Twice the row media of A321649 or A321650.
For mean instead of twice median we have A326839/A326840.
This is the conjugate version of A360005.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 is partition conjugation in terms of Heinz numbers.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A352491 gives n minus Heinz number of conjugate.
A363220 counts partitions with same median as conjugate.
Cf. A046682, A321648, A325040, A326567/A326568, A326848, A362617, A362618, A363223, A363224.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[If[n==1,0,2*Median[conj[prix[n]]]],{n,100}]

A361392 Number of integer partitions of n whose first differences have mean -1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 3, 2, 5, 4, 8, 7, 12, 12, 19, 19, 29, 31, 43, 48, 65, 73, 97, 110, 142, 164, 208, 240, 301, 350, 432, 504, 617, 719, 874, 1019, 1228, 1434, 1717, 2001, 2385, 2778, 3292, 3831, 4522, 5252, 6177, 7164, 8392, 9722, 11352, 13125, 15283, 17643

Offset: 0

Gus Wiseman, Mar 13 2023

nonn

These are partitions where the first part minus the last part is the number of parts minus 1.

			The a(3) = 1 through a(11) = 8 partitions:
  (21)  .  (32)   (321)  (43)    (422)   (54)     (442)    (65)
           (311)         (331)   (4211)  (432)    (4321)   (533)
                         (4111)          (4221)   (4411)   (4331)
                                         (4311)   (52111)  (4421)
                                         (51111)           (5222)
                                                           (52211)
                                                           (53111)
                                                           (611111)
For example, the partition y = (4,2,2,1) has first differences (-2,0,-1), with mean -1, so y is counted under a(9).

For mean 0 we have A032741.
The 0-appended version is A047993.
For any negative mean we have A144300.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A067538 counts partitions with integer mean, ranks A316413.
A326567/A326568 gives mean of prime indices, conjugate A326839/A326840.
A360614/A360615 gives mean of 0-appended first differences of prime indices.
Cf. A102627, A237363, A237832, A348551, A360688.

Table[Length[Select[IntegerPartitions[n],Mean[Differences[#]]==-1&]],{n,0,30}]

cp's OEIS Frontend

A326836 Heinz numbers of integer partitions whose maximum part divides their sum.

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A359912 Numbers whose prime indices do not have integer median.

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A349156 Number of integer partitions of n whose mean is not an integer.

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A348551 Heinz numbers of integer partitions whose mean is not an integer.

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A359913 Numbers whose multiset of prime factors has integer median.

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A326839 Numerator of A056239(n)/A061395(n) where A056239 is sum of prime indices and A061395 is maximum prime index.

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A361391 Number of strict integer partitions of n with non-integer mean.

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A363219 Twice the median of the conjugate of the integer partition with Heinz number n.

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A361392 Number of integer partitions of n whose first differences have mean -1.

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