A363127 Number of non-modes in the multiset of prime factors of n.

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, 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, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0

Offset: 1

Author

Gus Wiseman, May 16 2023

Keywords

• nonn

Comments

A non-mode in a multiset is an element that appears fewer times than at least one of the others. For example, the non-modes in {a,a,b,b,b,c,d,d,d} are {a,c}.

Examples

			The prime factorization of 13860 is 2*2*3*3*5*7*11, with non-modes {5,7,11}, so a(13860) = 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of first appearances converge to A088860.

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

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

The triangle for this rank statistic (number of non-modes) is A363126.

For non-co-modes instead of non-modes we have A363131, triangle A363130.

A027746 lists prime factors, A112798 indices, A124010 exponents.

A047966 counts uniform partitions, ranks A072774.

A363124 counts partitions with more than one non-mode, complement A363125.

Cf. A001221, A001222, A002865, A051903, A056239, A353863, A362616.

Programs

Maple

f:= proc(n) local F,m;
  F:= ifactors(n)[2][..,2];
  m:= max(F);
  nops(select(`<`,F,m))
end proc;
map(f, [$1..100]); # Robert Israel, Aug 01 2025

Mathematica

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
nmsi[ms_]:=Select[Union[ms],Count[ms,#]