cp's OEIS Frontend

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

%I A363127 #11 Aug 01 2025 02:13:50
%S A363127 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 A363127 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,
%U A363127 0,0,0,1,0,0,1,1,0,0,0,1,0,0,0,2,0,0,0
%N A363127 Number of non-modes in the multiset of prime factors of n.
%C A363127 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}.
%H A363127 Robert Israel, <a href="/A363127/b363127.txt">Table of n, a(n) for n = 1..10000</a>
%e A363127 The prime factorization of 13860 is 2*2*3*3*5*7*11, with non-modes {5,7,11}, so a(13860) = 3.
%p A363127 f:= proc(n) local F,m;
%p A363127   F:= ifactors(n)[2][..,2];
%p A363127   m:= max(F);
%p A363127   nops(select(`<`,F,m))
%p A363127 end proc;
%p A363127 map(f, [$1..100]); # _Robert Israel_, Aug 01 2025
%t A363127 prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A363127 nmsi[ms_]:=Select[Union[ms],Count[ms,#]<Max@@Length/@Split[ms]&];
%t A363127 Table[Length[nmsi[prifacs[n]]],{n,100}]
%Y A363127 Positions of first appearances converge to A088860.
%Y A363127 For modes instead of non-modes we have A362611, triangle A362614.
%Y A363127 For co-modes instead of non-modes we have A362613, triangle A362615.
%Y A363127 The triangle for this rank statistic (number of non-modes) is A363126.
%Y A363127 For non-co-modes instead of non-modes we have A363131, triangle A363130.
%Y A363127 A027746 lists prime factors, A112798 indices, A124010 exponents.
%Y A363127 A047966 counts uniform partitions, ranks A072774.
%Y A363127 A363124 counts partitions with more than one non-mode, complement A363125.
%Y A363127 Cf. A001221, A001222, A002865, A051903, A056239, A353863, A362616.
%K A363127 nonn
%O A363127 1,60
%A A363127 _Gus Wiseman_, May 16 2023