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A363741
Number of factorizations of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.
8
0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1
OFFSET
1,4
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 in {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).
Position of first appearance of n is: (1, 2, 4, 16, 64, 5832, 4096, ...).
EXAMPLE
The factorization 6*9*9*12 = 5832 has mean 9, median 9, and modes {9}, so it is counted under a(5832).
The a(n) factorizations for selected n:
2 4 16 64 5832 4096
2*2 4*4 8*8 18*18*18 64*64
2*2*2*2 4*4*4 6*9*9*12 8*8*8*8
2*2*2*2*2*2 3*6*6*6*9 16*16*16
2*3*3*3*3*3*3*4 4*4*4*4*4*4
2*2*2*2*2*2*2*2*2*2*2*2
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
modes[ms_]:=Select[Union[ms], Count[ms, #]>=Max@@Length/@Split[ms]&];
Table[Length[Select[facs[n], {Mean[#]}=={Median[#]}==modes[#]&]], {n, 100}]
CROSSREFS
For just (mean) = (median): A359909, see A240219, A359889, A359910, A359911.
The version for partitions is A363719, unequal A363720.
For unequal instead of equal we have A363742.
A000041 counts integer partitions.
A001055 counts factorizations, strict A045778, ordered A074206.
A089723 counts constant factorizations.
A316439 counts factorizations by length, A008284 partitions.
A326622 counts factorizations with integer mean, strict A328966.
A339846 counts even-length factorizations, A339890 odd-length.
Sequence in context: A341593 A288636 A089723 * A305253 A294336 A316782
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 26 2023
STATUS
approved