A328163 - OEIS

A328163

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

%I #7 Oct 09 2019 10:02:41

%S 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,

%T 130,73,170,66,251,78,287,137,364,119,522,135,590,236,759,190,1042,

%U 219,1175,425,1460,306,2006,347,2277,671,2780,471,3734,584,4197,1087

%N Number of integer partitions of n whose unsigned differences have a different GCD than 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(2) = 1 through a(12) = 15 partitions (A = 10, B = 11, C = 12):

%e (2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C)

%e (22) (33) (52) (44) (63) (55) (83) (66)

%e (42) (62) (72) (64) (92) (84)

%e (222) (422) (333) (73) (722) (93)

%e (2222) (522) (82) (5222) (A2)

%e (442) (444)

%e (622) (552)

%e (4222) (633)

%e (22222) (642)

%e (822)

%e (3333)

%e (4422)

%e (6222)

%e (42222)

%e (222222)

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

%Y The complement to these partitions is counted by A328164.

%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 0,5

%A _Gus Wiseman_, Oct 07 2019