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%I #13 Jan 10 2021 08:54:44
%S 1,1,1,1,1,2,1,3,2,4,4,5,4,9,9,10,12,14,16,20,23,29,34,38,41,51,60,66,
%T 78,89,103,119,137,157,180,201,229,261,298,338,379,431,486,547,618,
%U 694,783,876,986,1103,1241,1387,1551,1728,1932,2148,2395,2664,2963
%N Number of (necessarily strict) integer partitions of n with no two consecutive parts divisible.
%H Fausto A. C. Cariboni, <a href="/A328171/b328171.txt">Table of n, a(n) for n = 0..330</a>
%e The a(1) = 1 through a(15) = 10 partitions (A..F = 10..15):
%e 1 2 3 4 5 6 7 8 9 A B C D E F
%e 32 43 53 54 64 65 75 76 86 87
%e 52 72 73 74 543 85 95 96
%e 432 532 83 732 94 A4 B4
%e 92 A3 B3 D2
%e B2 653 654
%e 643 743 753
%e 652 752 852
%e 832 5432 A32
%e 6432
%t Table[Length[Select[IntegerPartitions[n],!MatchQ[#,{___,x_,y_,___}/;Divisible[x,y]]&]],{n,0,30}]
%Y The complement is counted by A328221.
%Y The Heinz numbers of these partitions are A328603.
%Y Partitions whose pairs of consecutive parts are relatively prime are A328172, with strict case A328188.
%Y Partitions with no pair of consecutive parts relatively prime are A328187, with strict case A328220.
%Y Numbers without consecutive divisible proper divisors are A328028.
%Y Cf. A000837, A018783, A328026, A328161, A328189, A328194, A328195.
%K nonn
%O 0,6
%A _Gus Wiseman_, Oct 11 2019