cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Gus Wiseman, Jul 16 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A326645.

Examples

			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)

Links

Crossrefs

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.

Programs

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

A326645 Heinz numbers of integer partitions whose mean and geometric mean are both integers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 46, 47, 49, 53, 57, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 183, 191, 193
Offset: 1

Views

Author

Gus Wiseman, Jul 16 2019

Keywords

Comments

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 A326641.

Examples

			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}
   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}
   41: {13}
   43: {14}
   46: {1,9}
   47: {15}
   49: {4,4}

Links

Crossrefs

Heinz numbers of partitions with integer mean are A316413.
Heinz numbers of partitions with integer geometric mean are A326623.
Heinz numbers of non-constant partitions with integer mean and geometric mean are A326646.
Partitions with integer mean and geometric mean are A326641.
Subsets with integer mean and geometric mean are A326643.
Strict partitions with integer mean and geometric mean are A326029.
Cf. A051293, A056239, A067538, A067539, A078175, A112798, A316413, A326623, A326644, A326646, A326647.

Programs

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

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

Views

Author

Gus Wiseman, Jul 16 2019

Keywords

Examples

			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}

Links

Crossrefs

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.

Programs

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

Extensions

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

Views

Author

Gus Wiseman, Jul 16 2019

Keywords

Examples

			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)

Links

Crossrefs

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.

Programs

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

Extensions

More terms from Jinyuan Wang, Jun 26 2020

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

Views

Author

Gus Wiseman, Jul 16 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A326646.

Examples

			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)

Links

Crossrefs

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.

Programs

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

Formula

a(n) = A326641(n) - A000005(n).
Showing 1-5 of 5 results.

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