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.

Previous Showing 11-20 of 32 results. Next

A379671 Array read by antidiagonals downward where A(n,k) is the number of finite sets of positive integers with sum n and product k.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Jan 01 2025

Keywords

Comments

Counts finite sets of positive integers by sum and product.

Examples

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  1   0   0   0   0   0   0   0   0   0   0   0
   n=2:  0   1   0   0   0   0   0   0   0   0   0   0
   n=3:  0   1   1   0   0   0   0   0   0   0   0   0
   n=4:  0   0   1   1   0   0   0   0   0   0   0   0
   n=5:  0   0   0   1   1   1   0   0   0   0   0   0
   n=6:  0   0   0   0   1   2   0   1   0   0   0   0
   n=7:  0   0   0   0   0   1   1   1   0   1   0   1
   n=8:  0   0   0   0   0   0   1   1   0   1   0   2
   n=9:  0   0   0   0   0   0   0   1   1   0   0   1
  n=10:  0   0   0   0   0   0   0   0   1   1   0   0
  n=11:  0   0   0   0   0   0   0   0   0   1   1   0
  n=12:  0   0   0   0   0   0   0   0   0   0   1   1
The A(8,12) = 2 sets are: {2,6}, {1,3,4}.
The A(14,40) = 2 sets are: {4,10}, {1,5,8}.
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 1
   n+k=3: 0 0 0
   n+k=4: 0 0 1 0
   n+k=5: 0 0 0 1 0
   n+k=6: 0 0 0 1 0 0
   n+k=7: 0 0 0 0 1 0 0
   n+k=8: 0 0 0 0 1 0 0 0
   n+k=9: 0 0 0 0 0 1 0 0 0
  n+k=10: 0 0 0 0 0 1 0 0 0 0
  n+k=11: 0 0 0 0 0 1 1 0 0 0 0
  n+k=12: 0 0 0 0 0 0 2 0 0 0 0 0
  n+k=13: 0 0 0 0 0 0 0 1 0 0 0 0 0
  n+k=14: 0 0 0 0 0 0 1 1 0 0 0 0 0 0
  n+k=15: 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0
  n+k=16: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
For example, antidiagonal n+k=11 counts the following sets:
  n=5: {2,3}
  n=6: {1,5}
so the 11th antidiagonal is: (0,0,0,0,0,1,1,0,0,0,0).

Crossrefs

Row sums are A000009 = strict partitions, non-strict A000041.
Column sums are 2*A045778 where A045778 = strict factorizations, non-strict A001055.
Antidiagonal sums are A379672, non-strict A379667 (zeros A379670).
Without ones we have A379678, antidiagonal sums A379679 (zeros A379680).
The non-strict version is A379666, without ones A379668.
A316439 counts factorizations by length, partitions A008284.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A002865, A003963, A028422, A069016, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.

Programs

  • Mathematica
    nn=12;
tt=Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

A379668 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n into parts > 1 with product k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Dec 31 2024

Keywords

Comments

This table counts finite multisets of positive integers > 1 by sum and product. Compare to the triangle A318950.

Examples

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  0   0   0   0   0   0   0   0   0   0   0   0
   n=2:  0   1   0   0   0   0   0   0   0   0   0   0
   n=3:  0   0   1   0   0   0   0   0   0   0   0   0
   n=4:  0   0   0   2   0   0   0   0   0   0   0   0
   n=5:  0   0   0   0   1   1   0   0   0   0   0   0
   n=6:  0   0   0   0   0   1   0   2   1   0   0   0
   n=7:  0   0   0   0   0   0   1   0   0   1   0   2
   n=8:  0   0   0   0   0   0   0   1   0   0   0   1
   n=9:  0   0   0   0   0   0   0   0   1   0   0   0
  n=10:  0   0   0   0   0   0   0   0   0   1   0   0
  n=11:  0   0   0   0   0   0   0   0   0   0   1   0
  n=12:  0   0   0   0   0   0   0   0   0   0   0   1
For example, the A(11,48) = 3 partitions are: (4,4,3), (4,3,2,2), (3,2,2,2,2).
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 0
   n+k=3: 0 0 0
   n+k=4: 0 0 1 0
   n+k=5: 0 0 0 0 0
   n+k=6: 0 0 0 1 0 0
   n+k=7: 0 0 0 0 0 0 0
   n+k=8: 0 0 0 0 2 0 0 0
   n+k=9: 0 0 0 0 0 0 0 0 0
  n+k=10: 0 0 0 0 0 1 0 0 0 0
  n+k=11: 0 0 0 0 0 1 0 0 0 0 0
  n+k=12: 0 0 0 0 0 0 1 0 0 0 0 0
  n+k=13: 0 0 0 0 0 0 0 0 0 0 0 0 0
  n+k=14: 0 0 0 0 0 0 2 1 0 0 0 0 0 0
  n+k=15: 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
  n+k=16: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
For example, antidiagonal n+k=14 counts the following partitions:
  n=6: (42), (222)
  n=7: (7)
so the 14th antidiagonal is: (0,0,0,0,0,0,2,1,0,0,0,0,0,0,0).

Crossrefs

Column sums are A001055 = factorizations, strict A045778.
Row sums are A002865 = partitions into parts > 1.
Take transpose and remove upper half (all zeros) to get A318950.
Allowing one gives A379666, antidiagonal sums A379667.
Antidiagonal sums are A379669, zeros A379670.
The strict case allowing ones is A379671, antidiagonal sums A379672.
The strict case is A379678, antidiagonal sums A379679 (zeros A379680).
A000041 counts integer partitions, strict A000009.
A316439 counts factorizations by length, A008284 partitions.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A028422, A069016, A319000, A319057, A319916, A325036, A325041, A325042, A326152, A326178, A379720.

Programs

  • Mathematica
    nn=15;
tt=Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

Formula

For n <= k we have A(n,k) = A318950(k,n).

A379667 Number of finite multisets of positive integers with sum + product = n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 11, 12, 13, 14, 15, 17, 19, 19, 20, 22, 26, 26, 29, 30, 31, 34, 35, 36, 38, 40, 43, 46, 48, 48, 50, 53, 55, 57, 61, 62, 66, 66, 69, 73, 75, 77, 79, 82, 83, 85, 89, 91, 94, 94, 95, 103, 106, 107, 111, 113, 116, 119, 121
Offset: 0

Views

Author

Gus Wiseman, Jan 03 2025

Keywords

Examples

			The partition (2,2,1) has sum + product equal to 5 + 4 = 9, so is counted under a(9).
The a(0) = 0 through a(8) = 5 partitions:
  .  ()  (1)  (11)  (2)    (21)    (3)      (31)      (4)
                    (111)  (1111)  (211)    (2111)    (22)
                                   (11111)  (111111)  (311)
                                                      (21111)
                                                      (1111111)

Crossrefs

Arrays counting multisets by sum and product:
- partitions: A379666, antidiagonal sums A379667 (this)
- partitions without ones: A379668, antidiagonal sums A379669 (zeros A379670)
- strict partitions: A379671, antidiagonal sums A379672
- strict partitions without ones: A379678, antidiagonal sums A379679 (zeros A379680)
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A326156, A326172, A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721
- different: A379736, ranks A379722, see A111133
A000041 counts integer partitions, strict A000009.
A025147 counts partitions into distinct parts > 1, non-strict A002865.
A316439 counts factorizations by length, partitions A008284.
A318950 counts factorizations by sum.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A003963, A028422, A069016, A319000, A319057, A319916, A325036, A325041, A326152, A326178, A379720.

Programs

  • Mathematica
    Table[Length[Select[Join@@Array[IntegerPartitions,n+1,0],Total[#]+Times@@#==n&]],{n,0,30}]

A379672 Number of finite sets of positive integers with sum + product = n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 3, 1, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 3, 1, 2, 4, 4, 4, 3, 2, 4, 3, 1, 5, 5, 2, 3, 4, 3, 3, 5, 5, 4, 2, 1, 5, 6, 3, 4, 4, 3, 4, 3, 2, 4, 6, 4, 5, 6, 3, 4, 5, 4, 4, 4, 5, 5, 2, 2, 6, 7, 4, 3, 5
Offset: 0

Views

Author

Gus Wiseman, Jan 03 2025

Keywords

Comments

Antidiagonal sums of A379671, starting with 0.
The only zeros are a(0) and a(3).

Examples

			The a(n) sets for n = 2, 11, 20, 35, 47, 60:
  {1}  {1,5}  {10}     {3,8}    {5,7}    {30}
       {2,3}  {2,6}    {1,17}   {1,23}   {1,5,9}
              {1,3,4}  {2,11}   {2,15}   {2,4,6}
                       {1,4,6}  {3,11}   {1,2,19}
                                {2,3,6}  {1,3,14}
                                         {1,4,11}

Crossrefs

Arrays counting multisets by sum and product:
- partitions: A379666, antidiagonal sums A379667
- partitions without ones: A379668, antidiagonal sums A379669 (zeros A379670)
- strict partitions: A379671, antidiagonal sums A379672 (this)
- strict partitions without ones: A379678, antidiagonal sums A379679 (zeros A379680)
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A326156, A326172, A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721
- different: A379736, ranks A379722, see A111133
A000041 counts integer partitions, strict A000009.
A025147 counts strict partitions into parts > 1, non-strict A002865.
A318950 counts factorizations by sum.
Cf. A003963, A028422, A096765, A319000, A319057, A319916, A325036, A326152, A326178.

Programs

  • Mathematica
    Table[Length[Select[Join@@Array[IntegerPartitions,n,0],UnsameQ@@#&&Total[#]+Times@@#==n&]],{n,0,30}]

Extensions

More terms from Jinyuan Wang, Jan 11 2025

A379679 Number of finite sets of positive integers > 1 with sum + product = n.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 0, 2, 2, 2, 2, 1, 1, 3, 0, 1, 4, 1, 1, 2, 2, 1, 2, 3, 2, 2, 0, 1, 4, 2, 1, 3, 1, 2, 2, 1, 1, 3, 3, 1, 4, 2, 1, 3, 2, 2, 2, 2, 3, 2, 0, 2, 4, 3, 1, 2, 3
Offset: 1

Views

Author

Gus Wiseman, Jan 03 2025

Keywords

Comments

Antidiagonal sums of A379678.

Examples

			The set {2,3,4,6} has sum 15 and product 144 so is counted under a(159).
The a(n) sets for n = 47, 89, 119, 159, 179, 239:
  {5,7}    {8,9}     {2,39}  {3,39}     {2,59}   {2,79}
  {2,15}   {2,29}    {3,29}  {4,31}     {3,44}   {3,59}
  {3,11}   {4,17}    {4,23}  {7,19}     {4,35}   {4,47}
  {2,3,6}  {5,14}    {5,19}  {9,15}     {5,29}   {5,39}
           {2,3,12}  {7,14}  {2,3,22}   {8,19}   {7,29}
                     {9,11}  {2,4,17}   {9,17}   {9,23}
                             {2,7,10}   {11,14}  {11,19}
                             {2,3,4,6}           {14,15}
                                                 {2,9,12}

Crossrefs

Arrays counting multisets by sum and product:
- partitions: A379666, antidiagonal sums A379667
- partitions without ones: A379668, antidiagonal sums A379669 (zeros A379670)
- strict partitions: A379671, antidiagonal sums A379672
- strict partitions without ones: A379678, antidiagonal sums A379679 (this) (zeros A379680)
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A326156, A326172, A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721
- different: A379736, ranks A379722, see A111133
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1, strict A025147.
A316439 counts factorizations by length, partitions A008284.
A318950 counts factorizations by sum.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A003963, A069016, A319000, A319057, A319916, A326152, A326178, A379720.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[Join@@Array[facs,n],UnsameQ@@#&&Total[#]+Times@@#==n&]],{n,100}]

A379669 Number of finite multisets of positive integers > 1 with sum + product = n.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 2, 2, 0, 1, 2, 4, 0, 3, 1, 1, 3, 1, 1, 2, 2, 3, 3, 2, 0, 2, 3, 2, 2, 4, 1, 4, 0, 3, 4, 2, 2, 2, 3, 1, 2, 4, 2, 3, 0, 1, 8, 3, 1, 4, 2, 3, 3, 2, 1, 3, 5, 1, 4, 3, 1, 4, 2, 7, 2, 3, 4, 3, 0, 2, 4, 6, 2, 4, 4
Offset: 0

Views

Author

Gus Wiseman, Jan 03 2025

Keywords

Examples

			The partition (3,2,2) has sum + product equal to 7 + 12 = 19, so is counted under a(19).
The a(n) partitions for n = 4, 8, 14, 24, 59:
  (2)  (4)    (7)      (12)       (9,5)
       (2,2)  (4,2)    (4,4)      (11,4)
              (2,2,2)  (4,2,2)    (14,3)
                       (2,2,2,2)  (19,2)
                                  (4,4,3)
                                  (11,2,2)
                                  (4,3,2,2)
                                  (3,2,2,2,2)

Crossrefs

Arrays counting multisets by sum and product:
- partitions: A379666, antidiagonal sums A379667
- partitions without ones: A379668, antidiagonal sums A379669 (this) (zeros A379670)
- strict partitions: A379671, antidiagonal sums A379672
- strict partitions without ones: A379678, antidiagonal sums A379679 (zeros A379680)
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A326156, A326172, A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721
- different: A379736, ranks A379722, see A111133
A000041 counts integer partitions, strict A000009.
A025147 counts strict partitions into parts > 1, non-strict A002865.
A318950 counts factorizations by sum.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A003963, A069016, A319000, A319057, A319916, A325036, A326152, A326178, A379720.

Programs

  • Mathematica
    Table[Length[Select[Select[Join@@Array[IntegerPartitions,n+1,0],FreeQ[#,1]&],Total[#]+Times@@#==n&]],{n,0,30}]

A379734 Number of integer partitions of n into parts > 1 whose product is a multiple of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 4, 3, 2, 1, 8, 1, 4, 8, 27, 1, 32, 1, 40, 24, 13, 1, 175, 56, 22, 188, 166, 1, 387, 1, 874, 166, 61, 410, 1833, 1, 98, 391, 3028, 1, 2704, 1, 1828, 5893, 239, 1, 16756, 3446, 9742, 1865, 5276, 1, 32927, 8179, 31643, 3840, 814, 1, 82958, 1
Offset: 1

Views

Author

Gus Wiseman, Jan 07 2025

Keywords

Comments

Allowing 1's gives A057568.

Examples

			The a(n) partitions for n = 6, 10, 12, 15, 22:
  (6)  (10)     (12)         (15)         (22)
       (5,3,2)  (6,6)        (6,5,4)      (11,6,5)
                (5,4,3)      (7,5,3)      (11,7,4)
                (6,4,2)      (10,3,2)     (11,8,3)
                (4,3,3,2)    (5,4,3,3)    (11,9,2)
                (5,3,2,2)    (5,5,3,2)    (11,4,4,3)
                (6,2,2,2)    (6,5,2,2)    (11,5,4,2)
                (3,3,2,2,2)  (5,3,3,2,2)  (11,6,3,2)
                                          (11,7,2,2)
                                          (11,3,3,3,2)
                                          (11,4,3,2,2)
                                          (11,5,2,2,2)
                                          (11,3,2,2,2,2)

Crossrefs

These partitions are ranked by the odd terms of A326149.
The strict version is A379735, allowing 1's A379733.
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1.
A379666 counts partitions by sum and product, without 1's A379668.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A069016, A318950, A319000, A319916, A324851, A325041, A326152, A379671, A379678.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=1, 1, 0), `if`(i<2, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, min(i, n-i), t/igcd(i, t)))))
    end:
a:= n-> `if`(isprime(n), 1, b(n$3)):
seq(a(n), n=1..70);  # Alois P. Heinz, Jan 07 2025
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Divisible[Times@@#,n]&]],{n,30}]

A379735 Number of strict integer partitions of n into parts > 1 whose product is a multiple of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 3, 4, 4, 1, 8, 1, 11, 9, 7, 1, 26, 7, 10, 18, 33, 1, 67, 1, 56, 37, 20, 69, 158, 1, 27, 70, 252, 1, 280, 1, 207, 402, 52, 1, 834, 133, 423, 226, 465, 1, 1132, 635, 1541, 388, 129, 1, 3377, 1, 171, 2891, 3561, 1674, 3154
Offset: 1

Views

Author

Gus Wiseman, Jan 07 2025

Keywords

Comments

These partitions are ranked by the odd squarefree terms of A326149.

Examples

			The a(n) partitions for n = 2, 9, 12, 15, 18, 20, 21:
  (2)  (9)    (12)     (15)      (18)       (20)         (21)
       (6,3)  (5,4,3)  (6,5,4)   (12,6)     (8,7,5)      (8,7,6)
              (6,4,2)  (7,5,3)   (9,5,4)    (10,6,4)     (9,7,5)
                       (10,3,2)  (9,6,3)    (10,8,2)     (11,7,3)
                                 (9,7,2)    (11,5,4)     (12,7,2)
                                 (6,5,4,3)  (12,5,3)     (14,4,3)
                                 (7,6,3,2)  (7,6,5,2)    (7,6,5,3)
                                 (9,4,3,2)  (8,5,4,3)    (9,7,3,2)
                                            (9,5,4,2)    (7,5,4,3,2)
                                            (10,5,3,2)
                                            (6,5,4,3,2)

Crossrefs

Allowing 1's gives A379733.
The non-strict version is A379734, allowing 1's A057568.
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1.
A379666 counts partitions by sum and product, without 1's A379668.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A069016, A319000, A319057, A319916, A324851, A326152, A379671, A379678.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&UnsameQ@@#&&Divisible[Times@@#,n]&]],{n,30}]

A322526 Number of integer partitions of n whose product of parts is a squarefree number.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 6, 8, 9, 10, 13, 15, 17, 21, 24, 27, 30, 36, 41, 46, 51, 57, 65, 73, 82, 90, 101, 109, 121, 134, 150, 164, 177, 193, 214, 232, 253, 278, 300, 324, 351, 386, 419, 452, 484, 521, 563, 610, 658, 706, 758, 809, 868, 938, 1006, 1071, 1140, 1220, 1307
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2018

Keywords

Comments

The parts of such a partition must also be squarefree and distinct except for any number of 1's.

Examples

			The a(8) = 9 partitions are (53), (71), (521), (611), (5111), (32111), (311111), (2111111), (11111111). Missing from this list are (8), (62), (44), (431), (422), (4211), (41111), (332), (3311), (3221), (2222), (22211), (221111).

Crossrefs

Partial sums of A322530.
Cf. A001055, A003963, A005117, A064573 , A073576, A096276, A301987, A302505, A319005, A319916, A320322, A322527, A322529, A322531.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],SquareFreeQ[Times@@#]&]],{n,30}]

A379320 Number of integer partitions of n whose product is a multiple of n + 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 2, 3, 0, 14, 0, 7, 15, 53, 0, 81, 0, 110, 61, 32, 0, 562, 170, 62, 621, 560, 0, 1400, 0, 3387, 569, 199, 1515, 7734, 0, 339, 1486, 13374, 0, 11926, 0, 8033, 27164, 913, 0, 85326, 15947, 47588, 8294, 25430, 0, 174779, 39748, 169009
Offset: 0

Views

Author

Gus Wiseman, Jan 18 2025

Keywords

Comments

Also the number of integer partitions of n containing 1 whose product is a multiple of n. Without requiring a 1 we get A057568.

Examples

			The a(5) = 1 through a(11) = 14 partitions:
  (3,2)  .  (4,2,1)    (3,3,2)    (5,4)      .  (8,3)
            (2,2,2,1)  (3,3,1,1)  (5,2,2)       (4,4,3)
                                  (5,2,1,1)     (6,3,2)
                                                (6,4,1)
                                                (4,3,2,2)
                                                (4,3,3,1)
                                                (6,2,2,1)
                                                (3,2,2,2,2)
                                                (3,3,2,2,1)
                                                (4,3,2,1,1)
                                                (6,2,1,1,1)
                                                (3,2,2,2,1,1)
                                                (4,3,1,1,1,1)
                                                (3,2,2,1,1,1,1)

Crossrefs

For n instead of n+1 we have A057568 (strict A379733), ranks A326149.
These partitions are ranked by A380217 = A379319/2 = (even case of A326149)/2.
The case of equality is A380218, see also A028422 = A001055 - 1 (ranks A325041).
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A036844, A069016, A318950, A319000, A319916, A324851, A325042, A326150, A326152, A326156, A379671, A379678, A379734.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n+1]&]],{n,0,30}]
  • PARI
    a(n) = my(nb=0); forpart(p=n, if (!(vecprod(Vec(p)) % (n+1)), nb++)); nb; \\ Michel Marcus, Jan 21 2025
Previous Showing 11-20 of 32 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.