A102627 -id:A102627 - cp's OEIS frontend

A067539 Number of partitions of n in which, if the number of parts is k, the product of the parts is the k-th power of some positive integer.

1, 2, 2, 3, 3, 4, 3, 4, 4, 8, 3, 8, 5, 7, 8, 8, 7, 9, 8, 17, 11, 11, 8, 16, 17, 17, 14, 18, 17, 26, 19, 24, 20, 30, 28, 32, 27, 37, 35, 48, 37, 45, 37, 51, 51, 58, 50, 64, 62, 83, 73, 84, 69, 91, 89, 101, 97, 116, 111, 136, 123, 142, 138, 160, 161, 181, 171, 205, 199, 231, 221

Offset: 1

Naohiro Nomoto, Jan 27 2002

a(n) is the number of integer partitions of n whose geometric mean is an integer. - Gus Wiseman, Jul 19 2019

			From _Gus Wiseman_, Jul 19 2019: (Start)
The a(1) = 1 through a(8) = 4 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (41)     (33)      (421)      (44)
                    (1111)  (11111)  (222)     (1111111)  (2222)
                                     (111111)             (11111111)
(End)

Wikipedia, Geometric mean

Partitions with integer average are A067538.
Subsets whose geometric mean is an integer are A326027.
The Heinz numbers of these partitions are A326623.
The strict case is A326625.
Cf. A000041, A102627, A320322, A326028, A326641.

Table[Length[Select[IntegerPartitions[n],IntegerQ[GeometricMean[#]]&]],{n,30}] (* Gus Wiseman, Jul 19 2019 *)

from math import prod
from sympy import integer_nthroot
from sympy.utilities.iterables import partitions
def A067539(n): return sum(1 for s,p in partitions(n,size=True) if integer_nthroot(prod(a**b for a, b in p.items()),s)[1]) # Chai Wah Wu, Sep 24 2023

Terms a(61) onwards from Max Alekseyev, Feb 06 2010

A326027 Number of nonempty subsets of {1..n} whose geometric mean is an integer.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 12, 19, 20, 21, 28, 29, 30, 31, 40, 41, 70, 71, 74, 75, 76, 77, 108, 123, 124, 211, 214, 215, 216, 217, 332, 333, 334, 335, 592, 593, 594, 595, 612, 613, 614, 615, 618, 639, 640, 641, 1160, 1183, 1324, 1325, 1328, 1329, 2176, 2177, 2196, 2197, 2198, 2199, 2414, 2415, 2416, 2443, 4000, 4001, 4002, 4003, 4006, 4007, 4008, 4009, 6626, 6627, 6628, 9753, 9756, 9757, 9758, 9759, 11136

Offset: 0

Gus Wiseman, Jul 14 2019

nonn

			The a(1) = 1 through a(9) = 19 subsets:
  {1}  {1}  {1}  {1}      {1}      {1}      {1}      {1}      {1}
       {2}  {2}  {2}      {2}      {2}      {2}      {2}      {2}
            {3}  {3}      {3}      {3}      {3}      {3}      {3}
                 {4}      {4}      {4}      {4}      {4}      {4}
                 {1,4}    {5}      {5}      {5}      {5}      {5}
                 {1,2,4}  {1,4}    {6}      {6}      {6}      {6}
                          {1,2,4}  {1,4}    {7}      {7}      {7}
                                   {1,2,4}  {1,4}    {8}      {8}
                                            {1,2,4}  {1,4}    {9}
                                                     {2,8}    {1,4}
                                                     {1,2,4}  {1,9}
                                                     {2,4,8}  {2,8}
                                                              {4,9}
                                                              {1,2,4}
                                                              {1,3,9}
                                                              {2,4,8}
                                                              {3,8,9}
                                                              {4,6,9}
                                                              {3,6,8,9}

Max Alekseyev, Table of n, a(n) for n = 0..255
Wikipedia, Geometric mean

First differences are A082553.
Partitions whose geometric mean is an integer are A067539.
Strict partitions whose geometric mean is an integer are A326625.
Subsets whose average is an integer are A051293.
Cf. A078174, A078175, A102627, A326567/A326568, A326622, A326623, A326624.

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

a(n) = A357413(n) + A357414(n). For a squarefree n, a(n) = a(n-1) + 1. - Max Alekseyev, Mar 01 2025

Terms a(57) onward from Max Alekseyev, Mar 01 2025

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[#]]]&]

A326621 Numbers n such that the average of the set of distinct prime indices of n is an integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 60, 61, 62, 63, 64, 67, 68, 71, 73, 78, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 94, 97, 100, 101, 103, 105

Offset: 1

Gus Wiseman, Jul 14 2019

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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose distinct parts have an integer average.

			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}
   10: {1,3}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}

Positions of 1's in A326620.

Cf. A051293, A056239, A067538, A078174, A078175, A102627, A112798, A326567/A326568, A326619/A326620, A326622.

Select[Range[2,100],IntegerQ[Mean[PrimePi/@First/@FactorInteger[#]]]&]

A326623 Heinz numbers of integer partitions whose geometric mean is an integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 42, 43, 46, 47, 49, 53, 57, 59, 61, 64, 67, 71, 73, 76, 79, 81, 83, 89, 97, 101, 103, 106, 107, 109, 113, 121, 125, 126, 127, 128, 131, 137, 139, 149, 151, 157, 161, 163, 167, 169

Offset: 1

Gus Wiseman, Jul 14 2019

nonn

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

			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}
   13: {6}
   14: {1,4}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Wikipedia, Geometric mean

The enumeration of these partitions by sum is given by A067539.
Heinz numbers of partitions with integer average are A316413.
The case without prime powers is A326624.
Subsets whose geometric mean is an integer are A326027.
Factorizations with integer geometric mean are A326028.
Cf. A001055, A078175, A102627, A326567/A326568, A326622, A326625.

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

A326669 Numbers k such that the average position of the ones in the binary expansion of k is an integer.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 14, 16, 17, 20, 21, 27, 28, 31, 32, 34, 35, 39, 40, 42, 49, 54, 56, 57, 62, 64, 65, 68, 70, 73, 78, 80, 84, 85, 93, 98, 99, 107, 108, 112, 114, 119, 124, 127, 128, 130, 133, 136, 140, 141, 146, 147, 155, 156, 160, 161, 167, 168, 170, 175

Offset: 1

Gus Wiseman, Jul 17 2019

These are numbers whose exponents in their representation as a sum of distinct powers of 2 have integer average.

			42 is in the sequence because 42 = 2^1 + 2^3 + 2^5 and the average of {1,3,5} is 3, an integer.

Cf. A000120, A051293, A070939, A102627, A291165, A291166, A316413, A326622, A326668, A326672, A326673.

Select[Range[100],IntegerQ[Mean[Join@@Position[IntegerDigits[#,2],1]]]&]

isok(m) = my(b=binary(m)); denominator(vecsum(Vec(select(x->(x==1), b, 1)))/hammingweight(m)) == 1; \\ Michel Marcus, Jul 02 2021

A363943 Mean of the multiset of prime indices of n, rounded down.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 2, 5, 1, 6, 2, 2, 1, 7, 1, 8, 1, 3, 3, 9, 1, 3, 3, 2, 2, 10, 2, 11, 1, 3, 4, 3, 1, 12, 4, 4, 1, 13, 2, 14, 2, 2, 5, 15, 1, 4, 2, 4, 2, 16, 1, 4, 1, 5, 5, 17, 1, 18, 6, 2, 1, 4, 2, 19, 3, 5, 2, 20, 1, 21, 6, 2, 3, 4, 3, 22, 1, 2, 7

Offset: 1

Gus Wiseman, Jun 29 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.

Extending the terminology introduced at A124943, this is the "low mean" of prime indices.

			The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 1.

Positions of first appearances are 1 and A000040.
Before rounding down we had A326567/A326568.
For mode instead of mean we have A363486, high A363487.
For low median instead of mean we have A363941, triangle A124943.
For high median instead of mean we have A363942, triangle A124944.
The high version is A363944, triangle A363946.
The triangle for this statistic (low mean) is A363945.
Positions of 1's are A363949(n) = 2*A344296(n), counted by A025065.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363947 ranks partitions with rounded mean 1, counted by A363948.
Cf. A102627, A327473, A327476, A327482, A348551, A359889, A363727, A363951.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
meandown[y_]:=If[Length[y]==0,0,Floor[Mean[y]]];
Table[meandown[prix[n]],{n,100}]

A326625 Number of strict integer partitions of n whose geometric mean is an integer.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 14 2019

nonn

			The a(63) = 9 partitions:
  (63)  (36,18,9)  (54,4,3,2)   (36,18,6,2,1)   (36,9,8,6,3,1)
        (48,12,3)  (27,24,8,4)  (18,16,12,9,8)
                   (32,18,9,4)
The initial terms count the following partitions:
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   5: (5)
   5: (4,1)
   6: (6)
   7: (7)
   7: (4,2,1)
   8: (8)
   9: (9)
  10: (10)
  10: (9,1)
  10: (8,2)
  11: (11)
  12: (12)
  13: (13)
  13: (9,4)
  13: (9,3,1)
  14: (14)
  14: (8,4,2)
  15: (15)
  15: (12,3)
  16: (16)

Wikipedia, Geometric mean

Partitions whose geometric mean is an integer are A067539.
Strict partitions whose average is an integer are A102627.
Cf. A078174, A078175, A326027, A326567/A326568, A326623, A326624.

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

A359897 Number of strict integer partitions of n whose parts have the same mean as median.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 4, 4, 7, 6, 6, 10, 7, 10, 13, 11, 9, 20, 10, 20, 18, 21, 12, 30, 24, 28, 27, 30, 15, 73, 16, 37, 43, 45, 67, 74, 19, 55, 71, 126, 21, 150, 22, 75, 225, 78, 24, 183, 126, 245, 192, 132, 27, 284, 244, 403, 303, 120, 30, 828

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(1) = 1 through a(9) = 7 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)  (5,4)
                          (4,1)  (5,1)    (5,2)  (6,2)  (6,3)
                                 (3,2,1)  (6,1)  (7,1)  (7,2)
                                                        (8,1)
                                                        (4,3,2)
                                                        (5,3,1)

The non-strict version is A240219, complement A359894, ranked by A359889.
The complement is counted by A359898.
The odd-length case is A359899, complement A359900.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A237984 counts partitions containing their mean, complement A327472.
A240850 counts strict partitions containing their mean, complement A240851.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
Cf. A065795, A066571, A067659, A082550, A102627, A135342, A316313, A327473, A327475, A328966, A359909.

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

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}]

cp's OEIS Frontend

A067539 Number of partitions of n in which, if the number of parts is k, the product of the parts is the k-th power of some positive integer.

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A326027 Number of nonempty subsets of {1..n} whose geometric mean is an integer.

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

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A326621 Numbers n such that the average of the set of distinct prime indices of n is an integer.

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A326623 Heinz numbers of integer partitions whose geometric mean is an integer.

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A326669 Numbers k such that the average position of the ones in the binary expansion of k is an integer.

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A363943 Mean of the multiset of prime indices of n, rounded down.

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A326625 Number of strict integer partitions of n whose geometric mean is an integer.

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