A326645 -id:A326645 - cp's OEIS frontend

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

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[#]]]&]

A326641 Number of integer partitions of n whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 2, 4, 3, 6, 2, 7, 2, 4, 5, 6, 2, 6, 2, 10, 6, 4, 2, 11, 4, 6, 5, 8, 2, 15, 2, 10, 6, 6, 8, 16, 2, 4, 8, 20, 2, 17, 2, 8, 17, 4, 2, 27, 9, 20, 8, 14, 2, 21, 10, 35, 10, 6, 2, 48, 2, 4, 41, 39, 12, 28, 2, 17, 10, 64, 2, 103, 2, 6, 23

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

The Heinz numbers of these partitions are given by A326645.

			The a(4) = 3 through a(10) = 6 partitions (A = 10):
  (4)     (5)      (6)       (7)        (8)         (9)          (A)
  (22)    (11111)  (33)      (1111111)  (44)        (333)        (55)
  (1111)           (222)                (2222)      (111111111)  (82)
                   (111111)             (11111111)               (91)
                                                                 (22222)
                                                                 (1111111111)

Wikipedia, Geometric mean

Partitions with integer mean are A067538.
Partitions with integer geometric mean are A067539.
Non-constant partitions with integer mean and geometric mean are A326642.
Subsets with integer mean and geometric mean are A326643.
Heinz numbers of partitions with integer mean and geometric mean are A326645.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A051293, A078175, A082553, A102627, A316413, A326027, A326623, A326644, A326646, A326647.

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

A326647 Number of factorizations of n into factors > 1 with integer average and integer geometric mean.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 16 2019

nonn

			The a(216) = 5 factorizations:
  (2*4*27)
  (3*3*24)
  (3*6*12)
  (6*6*6)
  (216)
The a(729) = 8 factorizations:
  (3*3*3*3*3*3)
  (3*3*81)
  (3*9*27)
  (3*243)
  (9*9*9)
  (9*81)
  (27*27)
  (729)

Wikipedia, Geometric mean

Positions of terms > 1 are the perfect powers A001597.
Factorizations with integer average are A326622.
Factorizations with integer geometric mean are A326028.
Partitions with integer average and geometric mean are A326641.
Cf. A001055, A067538, A067539, A322794, A326514, A326515, A326516, A326623, A326643, A326645.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,2,100}]

A326643 Number of subsets of {1..n} whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 30, 31, 32, 33, 34, 35, 41, 46, 47, 70, 71, 72, 73, 74, 102, 103, 104, 105, 143, 144, 145, 146, 151, 152, 153, 154, 155, 161, 162, 163, 244, 252, 280, 281, 282, 283, 409, 410, 416, 417, 418, 419

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

			The a(1) = 1 through a(12) = 16 subsets:
  {1}  {1}  {1}  {1}  {1}  {1}  {1}  {1}    {1}    {1}    {1}    {1}
       {2}  {2}  {2}  {2}  {2}  {2}  {2}    {2}    {2}    {2}    {2}
            {3}  {3}  {3}  {3}  {3}  {3}    {3}    {3}    {3}    {3}
                 {4}  {4}  {4}  {4}  {4}    {4}    {4}    {4}    {4}
                      {5}  {5}  {5}  {5}    {5}    {5}    {5}    {5}
                           {6}  {6}  {6}    {6}    {6}    {6}    {6}
                                {7}  {7}    {7}    {7}    {7}    {7}
                                     {8}    {8}    {8}    {8}    {8}
                                     {2,8}  {9}    {9}    {9}    {9}
                                            {1,9}  {10}   {10}   {10}
                                            {2,8}  {1,9}  {11}   {11}
                                                   {2,8}  {1,9}  {12}
                                                          {2,8}  {1,9}
                                                                 {2,8}
                                                                 {3,6,12}
                                                                 {3,4,9,12}

David Wasserman, Table of n, a(n) for n = 0..119
Wikipedia, Geometric mean

Partial sums of A326644.
Subsets whose geometric mean is an integer are A326027.
Subsets whose mean is an integer are A051293.
Partitions with integer mean and geometric mean are A326641.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A067538, A067539, A078175, A082553, A102627, A326623, A326625, A326645, A326646, A326647.

Table[Length[Select[Subsets[Range[n]],IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,0,10}]

More terms from David Wasserman, Aug 03 2019

A326029 Number of strict integer partitions of n whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 2, 3, 1, 3, 1, 1, 3, 6, 1, 3, 1, 2, 1, 1, 1, 3, 1, 6, 1, 5, 1, 2, 2, 2, 4, 3, 1, 9, 1, 1, 3, 1, 1, 4, 1, 4, 2, 6, 1, 6, 1, 3, 7, 4, 2, 5, 1, 10, 1, 3, 1, 9, 3

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

			The a(55) = 2 through a(60) = 9 partitions:
  (55)           (56)         (57)        (58)    (59)  (60)
  (27,16,9,2,1)  (24,18,8,6)  (49,7,1)    (49,9)        (54,6)
                              (27,25,5)   (50,8)        (48,12)
                              (27,18,12)                (27,24,9)
                                                        (27,24,6,2,1)
                                                        (36,12,9,2,1)
                                                        (36,9,6,4,3,2)
                                                        (24,18,9,6,2,1)
                                                        (27,16,9,4,3,1)

Wikipedia, Geometric mean

Partitions with integer mean and geometric mean are A326641.
Strict partitions with integer mean are A102627.
Strict partitions with integer geometric mean are A326625.
Non-constant partitions with integer mean and geometric mean are A326641.
Subsets with integer mean and geometric mean are A326643.
Heinz numbers of partitions with integer mean and geometric mean are A326645.
Cf. A051293, A067538, A067539, A078175, A082553, A316413, A326027, A326623, A326644, A326646, A326647.

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

More terms from Jinyuan Wang, Jun 26 2020

A326644 Number of subsets of {1..n} containing n whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 3, 1, 7, 1, 1, 1, 1, 1, 6, 5, 1, 23, 1, 1, 1, 1, 28, 1, 1, 1, 38, 1, 1, 1, 5, 1, 1, 1, 1, 6, 1, 1, 81, 8, 28, 1, 1, 1, 126, 1, 6, 1, 1, 1, 37, 1, 1, 6, 208, 1, 1, 1, 1, 1, 1, 1, 351, 1, 1, 381, 1, 1, 1, 1, 159, 605, 1, 1, 9, 1, 1, 1, 2, 1, 1223, 1, 1, 1, 1, 1, 805, 1, 113, 2, 5021, 1, 1, 1, 2, 1, 1, 1, 2630, 1, 1, 1, 54, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

			The a(1) = 1 through a(12) = 3 subsets:
  {1}  {2}  {3}  {4}  {5}  {6}  {7}  {8}    {9}    {10}  {11}  {12}
                                     {2,8}  {1,9}              {3,6,12}
                                                               {3,4,9,12}
The a(18) = 7 subsets:
  {18}
  {2,18}
  {8,18}
  {1,8,9,18}
  {2,3,8,9,18}
  {6,12,16,18}
  {2,3,4,9,12,18}

Wikipedia, Geometric mean

First differences of A326643.
Subsets whose mean is an integer are A051293.
Subsets whose geometric mean is an integer are A326027.
Partitions with integer mean and geometric mean are A326641.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A067538, A067539, A082550, A082553, A102627, A326028, A326623, A326625, A326645, A326647.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]],{n,0,10}]

More terms from David Wasserman, Aug 03 2019

A326666 Numbers k such that there exists a factorization of k into factors > 1 whose mean is not an integer but whose geometric mean is an integer.

Original entry on oeis.org

36, 64, 100, 144, 196, 216, 256, 324, 400, 484, 512, 576, 676, 784, 900, 1000, 1024, 1156, 1296, 1444, 1600, 1728, 1764, 1936, 2116, 2304, 2500, 2704, 2744, 2916, 3136, 3364, 3375, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 5832, 6084, 6400, 6724

Offset: 1

Gus Wiseman, Jul 17 2019

nonn

			36 has two such factorizations: (3*12) and (4*9).
The sequence of terms together with their prime indices begins:
    36: {1,1,2,2}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   144: {1,1,1,1,2,2}
   196: {1,1,4,4}
   216: {1,1,1,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   324: {1,1,2,2,2,2}
   400: {1,1,1,1,3,3}
   484: {1,1,5,5}
   512: {1,1,1,1,1,1,1,1,1}
   576: {1,1,1,1,1,1,2,2}
   676: {1,1,6,6}
   784: {1,1,1,1,4,4}
   900: {1,1,2,2,3,3}
  1000: {1,1,1,3,3,3}
  1024: {1,1,1,1,1,1,1,1,1,1}
  1156: {1,1,7,7}
  1296: {1,1,1,1,2,2,2,2}
  1444: {1,1,8,8}

Wikipedia, Geometric mean

A subsequence of A001597.
Factorizations with integer mean are A326622.
Factorizations with integer geometric mean are A326028.
Cf. A001055, A067538, A067539, A326027, A326516, A326623, A326641, A326643, A326645, A326647.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Length[Select[facs[#],!IntegerQ[Mean[#]]&&IntegerQ[GeometricMean[#]]&]]>1&]

A326646 Heinz numbers of non-constant integer partitions whose mean and geometric mean are both integers.

Original entry on oeis.org

46, 57, 183, 194, 228, 371, 393, 454, 515, 687, 742, 838, 1057, 1064, 1077, 1150, 1157, 1159, 1244, 1322, 1563, 1895, 2018, 2060, 2116, 2157, 2163, 2167, 2177, 2225, 2231, 2405, 2489, 2854, 2859, 3249, 3263, 3339, 3352, 3558, 3669, 3758, 3787, 3914, 4265, 4351

Offset: 1

Gus Wiseman, Jul 16 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A326642.

			The sequence of terms together with their prime indices begins:
    46: {1,9}
    57: {2,8}
   183: {2,18}
   194: {1,25}
   228: {1,1,2,8}
   371: {4,16}
   393: {2,32}
   454: {1,49}
   515: {3,27}
   687: {2,50}
   742: {1,4,16}
   838: {1,81}
  1057: {4,36}
  1064: {1,1,1,4,8}
  1077: {2,72}
  1150: {1,3,3,9}
  1157: {6,24}
  1159: {8,18}
  1244: {1,1,64}
  1322: {1,121}

Wikipedia, Geometric mean

Heinz numbers of partitions with integer mean and geometric mean are A326645.
Heinz numbers of partitions with integer mean are A316413.
Heinz numbers of partitions with integer geometric mean are A326623.
Non-constant partitions with integer mean and geometric mean are A326642.
Subsets with integer mean and geometric mean are A326643.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A051293, A067538, A067539, A078175, A082553, A102627, A326027, A326641, A326644, A326647.

A339452 Number of compositions (ordered partitions) of n into distinct parts such that the geometric mean of the parts is an integer.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 7, 1, 1, 5, 1, 1, 9, 7, 3, 1, 3, 1, 7, 11, 13, 1, 7, 1, 11, 35, 25, 31, 27, 5, 157, 1, 31, 131, 39, 31, 33, 37, 183, 179, 135, 157, 7, 265, 3, 871, 187, 865, 259, 879, 867, 179, 1593, 6073, 1593, 271, 5995, 149, 6661, 2411, 1509, 997, 1045, 5887

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(10) = 5 because we have [10], [9, 1], [1, 9], [8, 2] and [2, 8].

Eric W. Weisstein's World of Mathematics, Geometric Mean
Index entries for sequences related to compositions

For partitions we have A326625, non-strict A067539 (ranked by A326623).
The version for subsets is A326027.
For arithmetic mean we have A339175, non-strict A271654.
The non-strict case is counted by A357710, ranked by A357490.
A032020 counts strict compositions.
A067538 counts partitions with integer average.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A051293, A078174, A096199, A102627, A326622, A326624, A326028, A326641, A326645, A335405.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&IntegerQ[GeometricMean[#]]&]],{n,0,15}] (* Gus Wiseman, Oct 30 2022 *)

A326642 Number of non-constant integer partitions of n whose mean and geometric mean are both integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 0, 4, 2, 0, 0, 3, 1, 2, 1, 2, 0, 7, 0, 4, 2, 2, 4, 7, 0, 0, 4, 12, 0, 9, 0, 2, 11, 0, 0, 17, 6, 14, 4, 8, 0, 13, 6, 27, 6, 2, 0, 36, 0, 0, 35, 32, 8, 20, 0, 11, 6, 56, 0, 91, 0, 2, 17

Offset: 0

Gus Wiseman, Jul 16 2019

nonn

The Heinz numbers of these partitions are given by A326646.

			The a(30) = 7 partitions:
  (27,3)
  (24,6)
  (24,3,3)
  (16,8,2,2,2)
  (9,9,9,1,1,1)
  (8,8,8,2,2,2)
  (8,8,4,4,1,1,1,1,1,1)

Wikipedia, Geometric mean

Partitions with integer mean and geometric mean are A326641.
Heinz numbers of non-constant partitions with integer mean and geometric mean are A326646.
Non-constant partitions with integer geometric mean are A326624.
Cf. A051293, A067538, A067539, A102627, A316413, A326029, A326643, A326644, A326645, A326647.

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

a(n) = A326641(n) - A000005(n).

cp's OEIS Frontend

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

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A326641 Number of integer partitions of n whose mean and geometric mean are both integers.

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A326647 Number of factorizations of n into factors > 1 with integer average and integer geometric mean.

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A326643 Number of subsets of {1..n} whose mean and geometric mean are both integers.

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A326029 Number of strict integer partitions of n whose mean and geometric mean are both integers.

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A326644 Number of subsets of {1..n} containing n whose mean and geometric mean are both integers.

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A326666 Numbers k such that there exists a factorization of k into factors > 1 whose mean is not an integer but whose geometric mean is an integer.

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A326646 Heinz numbers of non-constant integer partitions whose mean and geometric mean are both integers.

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A339452 Number of compositions (ordered partitions) of n into distinct parts such that the geometric mean of the parts is an integer.

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