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%I #8 Oct 19 2019 14:45:38
%S 1,1,1,1,1,3,1,6,4,8,14,14,27,30,55,69,97,155,200,312,421,630,893,
%T 1260,1864,2600,3813,5395,7801,11196,15971,23126,32917,47514,67993,
%U 97670,140334,200913,289147,414119,595109,853751,1225086,1759405,2523151,3623984,5198759
%N Number of compositions of n with no part divisible by the next or the prior.
%H Andrew Howroyd, <a href="/A328508/b328508.txt">Table of n, a(n) for n = 0..1000</a>
%e The a(1) = 1 through a(11) = 14 compositions (A = 10, B = 11):
%e (1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)
%e (23) (25) (35) (27) (37) (29)
%e (32) (34) (53) (45) (46) (38)
%e (43) (323) (54) (64) (47)
%e (52) (72) (73) (56)
%e (232) (234) (235) (65)
%e (252) (253) (74)
%e (432) (325) (83)
%e (343) (92)
%e (352) (254)
%e (523) (272)
%e (532) (353)
%e (2323) (434)
%e (3232) (452)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{___,x_,y_,___}/;Divisible[y,x]||Divisible[x,y]]&]],{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(i%j && 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 only forbid parts to be divisible by the next, we get A328460.
%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 0,6
%A _Gus Wiseman_, Oct 17 2019
%E Terms a(26) and beyond from _Andrew Howroyd_, Oct 19 2019