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%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
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