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Number of non-co-modes in the prime factorization of n.
7

%I #6 May 18 2023 08:33:21

%S 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,

%T 0,0,0,0,0,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0,1,0,0,0,0,1,

%U 0,0,0,1,0,0,1,1,0,0,0,1,0,0,0,1,0,0,0

%N Number of non-co-modes in the prime factorization of n.

%C We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.

%e The prime factorization of 360 is 2*2*2*3*3*5, with non-co-modes {2,3}, so a(360) = 2.

%t prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];

%t ncomsi[ms_]:=Select[Union[ms],Count[ms,#]>Min@@Length/@Split[ms]&];

%t Table[Length[ncomsi[prifacs[n]]],{n,100}]

%Y Positions of terms > 0 are A059404.

%Y Positions of first appearances appear to converge to A228593.

%Y For modes instead of non-co-modes we have A362611, triangle A362614.

%Y For co-modes instead of non-co-modes we have A362613, triangle A362615.

%Y For parts instead of multiplicities we have A362983.

%Y For non-modes instead of non-co-modes we have A363127, triangle A363126.

%Y The triangle for this rank statistic (number of non-co-modes) is A363130.

%Y A027746 lists prime factors, A112798 indices, A124010 exponents.

%Y A047966 counts uniform partitions, ranks A072774.

%Y A363128 counts partitions with more than one non-mode, complement A363129.

%Y Cf. A001221, A001222, A002865, A051903, A056239, A098859, A237984, A327472, A353863, A362616.

%K nonn

%O 1

%A _Gus Wiseman_, May 18 2023