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

0,5

Comments

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

Links

Table of n, a(n) for n=0..57.

Example

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)

Mathematica

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

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.

Sequence in context: A161077 A339220 A161293 * A217916 A057923 A147763

Adjacent sequences: A328160 A328161 A328162 * A328164 A328165 A328166

Keyword

nonn

Author

Gus Wiseman, Oct 07 2019

Status

approved