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%I #4 Oct 16 2019 08:45:35
%S 0,0,1,2,4,5,10,12,20,26,38,51,73,92,126,166,219,283,369,470,604,763,
%T 968,1217,1534,1907,2376,2944,3640,4476,5501,6723,8212,9986,12130,
%U 14682,17748,21376,25717,30847,36959,44152,52688,62714,74557,88440,104775,123878
%N Number of integer partitions of n with at least one pair of consecutive divisible parts.
%C Includes all non-strict partitions.
%e The a(2) = 1 through a(8) = 20 partitions:
%e (11) (21) (22) (41) (33) (61) (44)
%e (111) (31) (221) (42) (322) (62)
%e (211) (311) (51) (331) (71)
%e (1111) (2111) (222) (421) (332)
%e (11111) (321) (511) (422)
%e (411) (2221) (431)
%e (2211) (3211) (521)
%e (3111) (4111) (611)
%e (21111) (22111) (2222)
%e (111111) (31111) (3221)
%e (211111) (3311)
%e (1111111) (4211)
%e (5111)
%e (22211)
%e (32111)
%e (41111)
%e (221111)
%e (311111)
%e (2111111)
%e (11111111)
%t Table[Length[Select[IntegerPartitions[n],MatchQ[#,{___,x_,y_,___}/;Divisible[x,y]]&]],{n,0,30}]
%Y The complement is counted by A328171.
%Y Partitions whose consecutive parts are relatively prime are A328172.
%Y Partitions with no pair of consecutive parts relatively prime are A328187.
%Y Numbers without consecutive divisible proper divisors are A328028.
%Y Cf. A000837, A018783, A328026, A328161, A328188, A328189, A328194, A328220.
%K nonn
%O 0,4
%A _Gus Wiseman_, Oct 15 2019