%I #7 Oct 19 2019 14:45:07

%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, k-1, r[i,k]=sum(j=1, k-i, if(j%i, r[j, k-i])))); 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