A362620 - OEIS

%I #12 Dec 15 2023 08:00:10

%S 12,20,24,28,40,44,45,48,52,56,60,63,68,72,76,80,84,88,90,92,96,99,

%T 104,112,116,117,120,124,126,132,135,136,140,144,148,152,153,156,160,

%U 164,168,171,172,175,176,180,184,188,189,192,198,200,204,207,208,212

%N Numbers whose greatest prime factor is not a mode, meaning it appears fewer times than some other.

%C First differs from A112769 in lacking 300.

%H Robert Israel, <a href="/A362620/b362620.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mode_(statistics)">Mode (statistics)</a>.

%e The prime factorization of 90 is 2*3*3*5, with modes {3} and maximum 5, so 90 is in the sequence.

%p filter:= proc(n) local F;

%p   F:= sort(ifactors(n)[2], (a,b) -> a[1]<b[1]);

%p   F[-1,2] < max(F[..,2])

%p end proc:

%p select(filter, [$2..300]); # _Robert Israel_, Dec 15 2023

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

%t Select[Range[2,100],FreeQ[Commonest[prifacs[#]],Max[prifacs[#]]]&]

%Y Partitions of this type are counted by A240302.

%Y The complement is A362619, counted by A171979.

%Y A027746 lists prime factors, A112798 indices, length A001222, sum A056239.

%Y A356862 ranks partitions with a unique mode, counted by A362608.

%Y A359178 ranks partitions with a unique co-mode, counted by A362610.

%Y A362605 ranks partitions with a more than one mode, counted by A362607.

%Y A362606 ranks partitions with a more than one co-mode, counted by A362609.

%Y A362611 counts modes in prime factorization, triangle version A362614.

%Y A362613 counts co-modes in prime factorization, triangle version A362615.

%Y A362621 ranks partitions with median equal to maximum, counted by A053263.

%Y Cf. A000040, A002865, A237824, A237984, A327473, A327476, A359908, A362616.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 11 2023