%I #11 Jun 05 2018 22:35:55
%S 1,2,3,4,7,7,15,16,27,30,56,56,100,105,157,188,287,303,470,524,724,
%T 850,1197,1339,1856,2135,2814,3305,4360,4951,6532,7561,9563,11195,
%U 14165,16328,20631,23866,29471,34320,42336,48672,59872,69139,83625,96911,117153
%N Number of reducible integer partitions of n.
%C A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is reducible if either m is of size 1 or gcd(m_1, ..., m_k) = 1 and the multiset {y_1, ..., y_k} is also reducible.
%e The a(6) = 7 reducible integer partitions are (6), (51), (411), (321), (3111), (21111), (111111). Missing from this list are (42), (33), (222), (2211).
%t ptnredQ[y_]:=Or[Length[y]==1,And[GCD@@y==1,ptnredQ[Sort[Length/@Split[y],Greater]]]];
%t Table[Length[Select[IntegerPartitions[n],ptnredQ]],{n,20}]
%Y Cf. A007916, A071625, A181819, A182850, A182857, A275870, A304465, A304660, A304687, A304818, A305564, A305565, A305566.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jun 05 2018