%I
%S 1,1,1,2,2,4,5,8,11,16,26,35,53,76,115,168,244,363,528,782,1144,1685,
%T 2474,3633,5347,7844,11539,16946,24919,36605,53782,79053,116142,
%U 170700,250800,368585,541610,795884,1169572,1718593,2525522,3711134,5453542,8013798,11776138
%N Number of compositions of n with no part divisible by the next.
%H Andrew Howroyd, <a href="/A328460/b328460.txt">Table of n, a(n) for n = 0..1000</a>
%e The a(1) = 1 through a(9) = 16 compositions:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (21) (31) (23) (42) (25) (35) (27)
%e (32) (51) (34) (53) (45)
%e (41) (231) (43) (62) (54)
%e (321) (52) (71) (63)
%e (61) (251) (72)
%e (232) (323) (81)
%e (421) (341) (234)
%e (431) (252)
%e (521) (342)
%e (2321) (351)
%e (423)
%e (432)
%e (531)
%e (621)
%e (3231)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{___,x_,y_,___}/;Divisible[y,x]]&]],{n,0,10}]
%o (PARI) seq(n)={my(r=matid(n)); for(k=1, n, for(i=1, k1, r[i,k]=sum(j=1, ki, if(j%i, r[j, ki])))); concat([1], vecsum(Col(r)))} \\ _Andrew Howroyd_, Oct 19 2019
%Y The case of partitions is A328171.
%Y If we also require no part to be divisible by the prior, we get A328508.
%Y Compositions with each part relatively prime to the next are A167606.
%Y Compositions with no part relatively prime to the next are A178470.
%Y Cf. A328026, A328028, A328161, A328172, A328189.
%K nonn
%O 0,4
%A _Gus Wiseman_, Oct 17 2019
%E Terms a(26) and beyond from _Andrew Howroyd_, Oct 19 2019
