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%I #7 Nov 01 2019 18:41:15
%S 1,1,2,2,3,3,4,5,6,8,9,13,13,22,23,30,36,50,54,77,85,113,135,170,194,
%T 256,303,369,440,545,640,792,931,1132,1347,1616,1909,2295,2712,3225,
%U 3799,4519,5310,6278,7365,8675,10170,11928,13940,16314,19046,22223,25856
%N Number of integer partitions of n with no two distinct consecutive parts divisible.
%e The a(1) = 1 through a(10) = 9 partitions (A = 10).
%e 1 2 3 4 5 6 7 8 9 A
%e 11 111 22 32 33 43 44 54 55
%e 1111 11111 222 52 53 72 64
%e 111111 322 332 333 73
%e 1111111 2222 432 433
%e 11111111 522 532
%e 3222 3322
%e 111111111 22222
%e 1111111111
%t Table[Length[Select[IntegerPartitions[n],!MatchQ[Union[#],{___,x_,y_,___}/;Divisible[y,x]]&]],{n,0,30}]
%Y The Heinz numbers of these partitions are given by A328674.
%Y The case involving all consecutive parts (not just distinct) is A328171.
%Y The version for relative primality instead of divisibility is A328187.
%Y Partitions with all consecutive parts divisible are A003238.
%Y Compositions without consecutive divisibilities are A328460.
%Y Cf. A305148, A316476, A318726, A328172, A328508, A328593, A328598, A328603.
%K nonn
%O 0,3
%A _Gus Wiseman_, Oct 29 2019
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