OFFSET
1,12
COMMENTS
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(8) = 1 through a(18) = 12 partitions:
3221 . 32221 . 4332 . 3222221 43332 5443 . 433332
5331 3322211 53331 6442 443331
322221 4222211 63321 7441 533322
422211 32222221 533331
33222211 543321
42222211 633321
52222111 733311
322222221
332222211
422222211
432222111
522222111
MATHEMATICA
modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&{Mean[#]}=={Median[#]}==modes[#]&]], {n, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 23 2023
STATUS
approved