A338554 Number of non-constant integer partitions of n whose parts have a common divisor > 1.

0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 5, 0, 9, 0, 13, 6, 18, 0, 33, 0, 40, 14, 54, 0, 87, 5, 99, 27, 133, 0, 211, 0, 226, 55, 295, 18, 443, 0, 488, 100, 637, 0, 912, 0, 1000, 198, 1253, 0, 1775, 13, 1988, 296, 2434, 0, 3358, 59, 3728, 489, 4563, 0, 6241, 0, 6840, 814

Offset

0,9

Links

Table of n, a(n) for n=0..63.

Formula

For n > 0, a(n) = A018783(n) - A000005(n) + 1.

Example

The a(6) = 2 through a(15) = 6 partitions (empty columns indicated by dots, A = 10, B = 11, C = 12):
  (42)  .  (62)   (63)  (64)    .  (84)     .  (86)      (96)
           (422)        (82)       (93)        (A4)      (A5)
                        (442)      (A2)        (C2)      (C3)
                        (622)      (633)       (644)     (663)
                        (4222)     (642)       (662)     (933)
                                   (822)       (842)     (6333)
                                   (4422)      (A22)
                                   (6222)      (4442)
                                   (42222)     (6422)
                                               (8222)
                                               (44222)
                                               (62222)
                                               (422222)

Mathematica

Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&GCD@@#>1&]], {n, 0, 30}]

Crossrefs

A046022 lists positions of zeros.

A082023(n) - A059841(n) is the 2-part version, n > 2.

A303280(n) - 1 is the strict case (n > 1).

A338552 lists the Heinz numbers of these partitions.

A338553 counts the complement, with Heinz numbers A338555.

A000005 counts constant partitions, with Heinz numbers A000961.

A000837 counts relatively prime partitions, with Heinz numbers A289509.

A018783 counts non-relatively prime partitions (ordered: A178472), with Heinz numbers A318978.

A282750 counts relatively prime partitions by sum and length.

Cf. A000010, A008284, A051424, A082024, A289508, A302698, A304712.

Sequence in context: A330602 A058241 A021827 * A317301 A131915 A078036

Adjacent sequences: A338551 A338552 A338553 * A338555 A338556 A338557

Keyword

nonn

Author

Gus Wiseman, Nov 07 2020

Status

approved