A082553 -id:A082553 - cp's OEIS frontend

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

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

A326028 Number of factorizations of n into factors > 1 with 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, 5, 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, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Jul 15 2019

nonn

First differs from A294336 and A316782 at a(36) = 5.

			The a(4) = 2 through a(36) = 5 factorizations (showing only the cases where n is a perfect power).
  (4)    (8)      (9)    (16)       (25)   (27)     (32)         (36)
  (2*2)  (2*2*2)  (3*3)  (2*8)      (5*5)  (3*3*3)  (2*2*2*2*2)  (4*9)
                         (4*4)                                   (6*6)
                         (2*2*2*2)                               (2*18)
                                                                 (3*12)

Antti Karttunen, Table of n, a(n) for n = 1..100000
Wikipedia, Geometric mean
Index entries for sequences computed from exponents in factorization of n

Positions of terms > 1 are the perfect powers A001597.
Partitions with integer geometric mean are A067539.
Subsets with integer geometric mean are A326027.
Factorizations with integer average and geometric mean are A326647.
Cf. A001055, A082553, A322794, A326514, A326515, A326516, A326622, A326623, A326624, A326625.

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[GeometricMean[#]]&]],{n,2,100}]

A326028(n, m=n, facmul=1, facnum=0) = if(1==n,facnum>0 && ispower(facmul,facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326028(n/d, d, facmul*d, facnum+1))); (s)); \\ Antti Karttunen, Nov 10 2024

a(2^n) = A067538(n).

a(89) onwards from Antti Karttunen, Nov 10 2024

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}]

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

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.

cp's OEIS Frontend

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

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A326028 Number of factorizations of n into factors > 1 with integer geometric mean.

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

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

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