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

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

0,8

Comments

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]

Links

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

Example

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)

Mathematica

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

Prog

(PARI)
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

Crossrefs

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.

Sequence in context: A055212 A360661 A028422 * A064577 A371729 A322435

Adjacent sequences: A380215 A380216 A380217 * A380219 A380220 A380221

Keyword

nonn

Author

Gus Wiseman, Jan 21 2025

Extensions

More terms from Antti Karttunen, Jan 28 2025

Status

approved