%I #13 Dec 26 2020 16:06:54
%S 1,1,1,1,2,1,4,1,5,3,8,1,14,1,16,9,22,3,38,4,46,19,58,9,94,18,106,41,
%T 144,28,221,37,246,92,318,87,465,95,530,198,693,169,963,220,1108,424,
%U 1383,381,1899,492,2216,815,2732,799,3644,1041,4231,1585,5194,1608
%N Number of integer partitions of n with no pair of consecutive parts relatively prime.
%H Fausto A. C. Cariboni, <a href="/A328187/b328187.txt">Table of n, a(n) for n = 0..300</a>
%e The a(1) = 1 through a(15) = 9 partitions (A..F = 10..15):
%e 1 2 3 4 5 6 7 8 9 A B C D E F
%e 22 33 44 63 55 66 77 96
%e 42 62 333 64 84 86 A5
%e 222 422 82 93 A4 C3
%e 2222 442 A2 C2 555
%e 622 444 644 663
%e 4222 633 662 933
%e 22222 642 842 6333
%e 822 A22 33333
%e 3333 4442
%e 4422 6422
%e 6222 8222
%e 42222 44222
%e 222222 62222
%e 422222
%e 2222222
%t Table[Length[Select[IntegerPartitions[n],!MatchQ[#,{___,x_,y_,___}/;GCD[x,y]==1]&]],{n,0,30}]
%Y The Heinz numbers of these partitions are given by A328336.
%Y The case of compositions is A178470.
%Y The strict case is A328220.
%Y Partitions with all pairs of consecutive parts relatively prime are A328172.
%Y Cf. A000837, A018783, A328028, A328170, A328171, A328188.
%K nonn
%O 0,5
%A _Gus Wiseman_, Oct 12 2019
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