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A275870
Number of collapsible integer partitions of n.
95
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
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).
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 11 2016
STATUS
approved