A328164 Number of integer partitions of n whose unsigned differences have the same GCD as the GCD of their parts all minus 1.

1, 1, 1, 2, 3, 6, 7, 13, 17, 25, 33, 51, 62, 92, 116, 160, 203, 281, 341, 469, 572, 754, 929, 1221, 1466, 1912, 2306, 2937, 3548, 4499, 5353, 6764, 8062, 10006, 11946, 14764, 17455, 21502, 25425, 30949, 36579, 44393, 52132, 63042, 74000, 88709, 104098, 124448

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(1) = 1 through a(8) = 17 partitions:
  (1)  (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (61)       (71)
                    (1111)  (221)    (411)     (322)      (332)
                            (311)    (2211)    (331)      (431)
                            (2111)   (3111)    (421)      (521)
                            (11111)  (21111)   (511)      (611)
                                     (111111)  (2221)     (3221)
                                               (3211)     (3311)
                                               (4111)     (4211)
                                               (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
		

Crossrefs

The complement to these partitions is counted by A328163.

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}]