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%I #6 Sep 17 2018 08:35:07
%S 1,0,1,1,1,1,0,1,2,1,2,1,0,1,4,1,4,1,0,1,6,1,1,7,2,0,1,11,1,10,2,1,1,
%T 0,1,17,1,17,4,0,1,23,2,1,1,26,4,1,0,1,37,1,36,6,2,1,0,1,53,1,53,7,2,
%U 1,0,1,70,4,1,1,77,11,0,1,103,1,103,10,4,2,1
%N Irregular triangle where T(n,k) is the number of strict integer partitions of n with GCD equal to the k-th divisor of n.
%F T(n,k) = A078374(n/A027750(n,k)).
%e Triangle begins:
%e 1
%e 0 1
%e 1 1
%e 1 0 1
%e 2 1
%e 2 1 0 1
%e 4 1
%e 4 1 0 1
%e 6 1 1
%e 7 2 0 1
%e 11 1
%e 10 2 1 1 0 1
%e 17 1
%e 17 4 0 1
%e 23 2 1 1
%e 26 4 1 0 1
%e 37 1
%e 36 6 2 1 0 1
%e 53 1
%e 53 7 2 1 0 1
%t Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,GCD@@#==k]&]],{n,20},{k,Divisors[n]}]
%Y A regular version is A303138. Row lengths are A000005. Row sums are A000009. First column is A078374.
%Y Cf. A000837, A027750, A289508, A289509, A303140, A303280, A319299, A319301.
%K nonn,tabf
%O 1,9
%A _Gus Wiseman_, Sep 16 2018