A380216 -id:A380216 - cp's OEIS frontend

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

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

Gus Wiseman, Jan 18 2025

nonn

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

			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)

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.

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n+1]&]],{n,0,30}]

a(n) = my(nb=0); forpart(p=n, if (!(vecprod(Vec(p)) % (n+1)), nb++)); nb; \\ Michel Marcus, Jan 21 2025

A380218 Number of integer partitions of n with product n+1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 4, 0, 3, 0, 3, 1, 1, 0, 6, 1, 1, 2, 3, 0, 4, 0, 6, 1, 1, 1, 8, 0, 1, 1, 6, 0, 4, 0, 3, 3, 1, 0, 11, 1, 3, 1, 3, 0, 6, 1, 6, 1, 1, 0, 10, 0, 1, 3, 10, 1, 4, 0, 3, 1, 4, 0, 15, 0, 1, 3, 3, 1, 4, 0, 11, 4, 1, 0, 10, 1, 1, 1, 6, 0, 10, 1, 3, 1, 1, 1, 18, 0, 3, 3, 8, 0, 4, 0, 6, 4, 1

Offset: 0

Gus Wiseman, Jan 21 2025

nonn

For n instead of n+1 we have 0 followed by A001055.

Also the number of integer factorizations of n with sum < n. [When the sequence is interpreted as having offset 1 instead of 0. Clarified by Antti Karttunen, Jan 28 2025]

			The a(5) = 1 through a(15) = 4 partitions with product n+1:
  32  .  421   3311  5211  .  62111    .  721111  53111111  8211111
         2221                 431111                        441111111
                              3221111                       4221111111
                                                            22221111111
The a(1) = 1 through a(12) = 3 factorizations with sum < n:
  ()  .  .  .  .  (2*3)  .  (2*4)    (3*3)  (2*5)  .  (2*6)
                            (2*2*2)                   (3*4)
                                                      (2*2*3)

Antti Karttunen, Table of n, a(n) for n = 0..20000

Same as A028422 = A001055-1 except initial terms.
These partitions are ranked by A325041.
The version for divisibility instead of equality is A379320.
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 (strict A379733), ranks A326149, see A379319, A380217.
- 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, A069016, A318950, A319000, A319916, A325042, A326152, A326156, A379734.

Table[Length[Select[IntegerPartitions[n],SameQ[Times@@#,n+1]&]],{n,0,30}]

A380218off1(n, m=n, e=n) = if(1==n, (e>0), sumdiv(n, d, if((d>1)&&(d<=m),  A380218off1(n/d, d, e-d))));
A380218off0(n) = A380218off1(1+n); \\ Antti Karttunen, Jan 28 2025

More terms from Antti Karttunen, Jan 28 2025

A380219 Number of integer partitions of n whose product is a proper multiple of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 0, 18, 0, 9, 21, 75, 0, 109, 0, 146, 83, 43, 0, 730, 224, 82, 806, 722, 0, 1782, 0, 4254, 733, 258, 1923, 9558, 0, 435, 1875, 16395, 0, 14625, 0, 9857, 33053, 1150, 0, 102070, 19391, 57326, 10157, 30702, 0, 207699, 47925, 200645

Offset: 1

Gus Wiseman, Jan 21 2025

nonn

			The partition y = (4,3,3,2) has product 72, which is a multiple of 12, so y is counted under a(12).
The a(8) = 3 through a(14) = 9 partitions:
  (44)    (63)    (532)   .  (66)       .  (743)
  (422)   (333)   (541)      (543)         (752)
  (2222)  (3321)  (5221)     (642)         (761)
                             (831)         (7322)
                             (4332)        (7421)
                             (4431)        (72221)
                             (5322)        (73211)
                             (6222)        (74111)
                             (6321)        (722111)
                             (6411)
                             (33222)
                             (43221)
                             (43311)
                             (62211)
                             (322221)
                             (332211)
                             (432111)
                             (3222111)

The non-proper version is A057568, case of equality A001055.
The case of strict partitions is A379733 - 1.
The case of partitions without 1's is A379734 - 1.
These partitions are ranked by A380216.
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568 (strict A379733), ranks A326149, see A379319, A380217.
- 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, A028422, A318950, A319000, A319916, A326152, A326156, A379320, A379671, A379735, A380218, A380221.

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

a(n) = my(nb=0); forpart(p=n, my(vp=vecprod(Vec(p))); if (!(vp%n) && (vp>n), nb++)); nb; \\ Michel Marcus, Jan 22 2025

a(n) = A057568(n) - A001055(n).

A380221 Number of strict integer partitions of n containing 1 whose product of parts is a multiple of n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 3, 3, 0, 4, 0, 9, 6, 4, 0, 22, 5, 6, 15, 28, 0, 54, 0, 49, 30, 14, 57, 134, 0, 22, 58, 219, 0, 242, 0, 180, 349, 44, 0, 722, 113, 369, 196, 404, 0, 994, 556, 1363, 338, 111, 0, 3016, 0, 150, 2569, 3150, 1485, 2815, 0

Offset: 1

Gus Wiseman, Jan 22 2025

nonn

Also the number of strict integer partitions of n - 1 not containing 1 whose product of parts is a multiple of n. These are strict integer factorizations of multiples of n summing to n - 1.

			The a(6) = 1 through a(16) = 3 partitions:
  (3,2,1) . . . (5,4,1) . (8,3,1)   . (7,6,1)   (9,5,1)     (8,4,3,1)
                          (6,3,2,1)   (7,4,2,1) (6,5,3,1)   (8,5,2,1)
                                                (5,4,3,2,1) (6,4,3,2,1)

Positions of 0 after 9 appear to be the prime numbers A000040.
The non-strict version is A379320 shifted right, ranks A380217 = A379319/2.
Not requiring 1 gives A379733.
For n instead of n+1 we have A379735 shifted left, non-strict A379734.
Partitions of this type are ranked by A379845.
The case of equality for non-strict partitions is A380218 shifted left.
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, A069016, A319000, A319916, A325042, A326150, A326152, A326156, A379671, A379678.

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

A380343 Number of strict integer partitions of n whose product of parts is a multiple of n + 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 3, 5, 5, 0, 8, 0, 15, 11, 8, 0, 42, 8, 12, 26, 49, 0, 100, 0, 90, 56, 27, 105, 246, 0, 41, 108, 414, 0, 450, 0, 332, 651, 81, 0, 1341, 210, 693, 366, 754, 0, 1869, 1044, 2579, 634, 206, 0, 5695, 0, 278, 4850, 5927, 2802

Offset: 0

Gus Wiseman, Jan 22 2025

nonn

			The a(5) = 1 through a(17) = 8 partitions (A=10, C=12):
  32  .  421  .  54  .  83   .  76    95    843   .  98
                        632     742   653   852      863
                        641     7321  A31   861      962
                                      5432  6432     C32
                                      6521  8421     7631
                                                     9431
                                                     9521
                                                     65321

The non-strict version is A379320, ranked by A380217 = A379319/2.
For n instead of n+1 we have A379733, non-strict A057568.
The case of equality for non-strict partitions is A380218.
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, A069016, A319000, A319916, A325042, A326152, A326156, A379671, A379734.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@#,n+1]&]],{n,0,30}]

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

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A380218 Number of integer partitions of n with product n+1.

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A380219 Number of integer partitions of n whose product is a proper multiple of n.

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A380221 Number of strict integer partitions of n containing 1 whose product of parts is a multiple of n.

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A380343 Number of strict integer partitions of n whose product of parts is a multiple of n + 1.

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