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

Author

Gus Wiseman, Nov 07 2020

Keywords

• nonn

Examples

			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)
		

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.

Programs

Mathematica

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

Formula

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