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 A275870 Number of collapsible integer partitions of n. 34
 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, 94, 2, 297, 2, 202, 16, 21, 14, 1488, 2, 23, 18, 776, 2, 610, 2, 186, 148, 27, 2, 6978, 9, 319 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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). LINKS Gus Wiseman, Hasse diagram for the case n=16 with full detail FORMULA a(2^n)=A002577(n+1). MATHEMATICA 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, #]]&]]]]; repenum[n_]:=Length[Select[IntegerPartitions[n], MemberQ[repcaps[#], {n}]&]]; Table[repenum[n], {n, 1, 32}](* Gus Wiseman, Aug 11 2016 *) CROSSREFS Cf. A002577, A145515, A018818. Sequence in context: A057767 A207329 A122977 * A321721 A003980 A286369 Adjacent sequences:  A275867 A275868 A275869 * A275871 A275872 A275873 KEYWORD nonn AUTHOR Gus Wiseman, Aug 11 2016 STATUS approved

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Last modified June 24 18:46 EDT 2021. Contains 345419 sequences. (Running on oeis4.)