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%I #7 Aug 11 2016 23:37:38
%S 1,2,2,4,2,7,2,10,5,9,2,34,2,11,10,36,2,64,2,60,12,15,2,320,7,17,23,
%T 94,2,297,2,202,16,21,14,1488,2,23,18,776,2,610,2,186,148,27,2,6978,9,
%U 319
%N Number of collapsible integer partitions of n.
%C If a collapse is a joining of some number of equal parts of an integer partition p, we say p is collapsible if by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (32211111)->(332211)->(33222)->(6222)->(66)->(n) which shows that (32211111) is a collapsible partition of n=twelve.
%C One can show that if n is a power of a prime, then a partition of n is collapsible iff its parts are all divisors of n; so this sequence shares many terms with A145515 (number of partitions of k^n into powers of k) and A018818 (number of partitions of n into divisors of n).
%H Gus Wiseman, <a href="/A275870/a275870.png">The first 16 terms illustrated (together with the partial order induced by the collapsing relation)</a>
%H Gus Wiseman, <a href="/A275870/a275870_1.png">Hasse diagram for the case n=16 with full detail</a>
%F a(2^n)=A002577(n+1).
%t repcaps[q_List]:=repcaps[q]=Union[{q},If[UnsameQ@@q,{},Union@@repcaps/@Union[Sort[Append[Drop[q,#],Plus@@Take[q,#]],Greater]&/@Select[Tuples[Range[Length[q]],2],And[Less@@#,SameQ@@Take[q,#]]&]]]];
%t repenum[n_]:=Length[Select[IntegerPartitions[n],MemberQ[repcaps[#],{n}]&]];
%t Table[repenum[n],{n,1,32}](* _Gus Wiseman_, Aug 11 2016 *)
%Y Cf. A002577, A145515, A018818.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 11 2016
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