A328164 - OEIS

%I #5 Oct 09 2019 10:02:54

%S 1,1,1,2,3,6,7,13,17,25,33,51,62,92,116,160,203,281,341,469,572,754,

%T 929,1221,1466,1912,2306,2937,3548,4499,5353,6764,8062,10006,11946,

%U 14764,17455,21502,25425,30949,36579,44393,52132,63042,74000,88709,104098,124448

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

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

%e The a(1) = 1 through a(8) = 17 partitions:

%e   (1)  (11)  (21)   (31)    (32)     (51)      (43)       (53)

%e              (111)  (211)   (41)     (321)     (61)       (71)

%e                     (1111)  (221)    (411)     (322)      (332)

%e                             (311)    (2211)    (331)      (431)

%e                             (2111)   (3111)    (421)      (521)

%e                             (11111)  (21111)   (511)      (611)

%e                                      (111111)  (2221)     (3221)

%e                                                (3211)     (3311)

%e                                                (4111)     (4211)

%e                                                (22111)    (5111)

%e                                                (31111)    (22211)

%e                                                (211111)   (32111)

%e                                                (1111111)  (41111)

%e                                                           (221111)

%e                                                           (311111)

%e                                                           (2111111)

%e                                                           (11111111)

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

%Y The complement to these partitions is counted by A328163.

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

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

%Y Partitions whose parts minus 1 are relatively prime are A328170.

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Oct 07 2019