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A275870
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Number of collapsible integer partitions of n.
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95
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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