A328163 Number of integer partitions of n whose unsigned differences have a different GCD than the GCD of their parts all minus 1.

0, 0, 1, 1, 2, 1, 4, 2, 5, 5, 9, 5, 15, 9, 19, 16, 28, 16, 44, 21, 55, 38, 73, 34, 109, 46, 130, 73, 170, 66, 251, 78, 287, 137, 364, 119, 522, 135, 590, 236, 759, 190, 1042, 219, 1175, 425, 1460, 306, 2006, 347, 2277, 671, 2780, 471, 3734, 584, 4197, 1087

Offset: 0

Author

Gus Wiseman, Oct 07 2019

Keywords

• nonn

Comments

Zeros are ignored when computing GCD, and the empty set has GCD 0.

Examples

			The a(2) = 1 through a(12) = 15 partitions (A = 10, B = 11, C = 12):
  (2)  (3)  (4)   (5)  (6)    (7)   (8)     (9)    (A)      (B)     (C)
            (22)       (33)   (52)  (44)    (63)   (55)     (83)    (66)
                       (42)         (62)    (72)   (64)     (92)    (84)
                       (222)        (422)   (333)  (73)     (722)   (93)
                                    (2222)  (522)  (82)     (5222)  (A2)
                                                   (442)            (444)
                                                   (622)            (552)
                                                   (4222)           (633)
                                                   (22222)          (642)
                                                                    (822)
                                                                    (3333)
                                                                    (4422)
                                                                    (6222)
                                                                    (42222)
                                                                    (222222)
		

Crossrefs

The complement to these partitions is counted by A328164.

The GCD of the divisors of n all minus 1 is A258409(n).

The GCD of the prime indices of n all minus 1 is A328167(n).

Partitions whose parts minus 1 are relatively prime are A328170.

Cf. A000837, A018783, A175342, A279945, A289508, A328168, A328169.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],GCD@@Differences[#]!=GCD@@(#-1)&]],{n,0,30}]