%I #8 Dec 26 2020 03:31:10

%S 1,1,1,1,1,1,2,1,2,2,3,1,5,1,5,4,6,3,10,3,11,7,12,3,19,5,18,12,23,9,

%T 36,11,33,21,40,20,58,19,58,35,70,31,98,36,101,65,112,56,155,64,164,

%U 97,188,88,250,112,256,157,293,145,392,163,399,241,461,242

%N Number of strict integer partitions of n with no pair of consecutive parts relatively prime.

%H Fausto A. C. Cariboni, <a href="/A328220/b328220.txt">Table of n, a(n) for n = 0..500</a>

%e The a(2) = 1 through a(20) = 11 partitions (A..K = 10..20):

%e 2 3 4 5 6 7 8 9 A B C D E F G H I J K

%e 42 62 63 64 84 86 96 A6 863 A8 964 C8

%e 82 93 A4 A5 C4 962 C6 A63 E6

%e A2 C2 C3 E2 E4 F5

%e 642 842 862 F3 G4

%e A42 G2 I2

%e 864 A64

%e 963 A82

%e A62 C62

%e C42 E42

%e 8642

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[#,{___,x_,y_,___}/;GCD[x,y]==1]&]],{n,0,30}]

%Y The non-strict case is A328187.

%Y Partitions with all consecutive parts relatively prime are A328172, with strict case A328188.

%Y Strict partitions with relatively prime parts are A078374.

%Y Partitions with no consecutive divisibilities are A328171.

%Y Cf. A000837, A018783, A070211, A289509, A318980, A325545, A328170, A328221, A328335, A328336.

%K nonn

%O 0,7

%A _Gus Wiseman_, Oct 14 2019