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Number of modes in the prime factorization of n.
56

%I #15 May 09 2023 00:27:47

%S 0,1,1,1,1,2,1,1,1,2,1,1,1,2,2,1,1,1,1,1,2,2,1,1,1,2,1,1,1,3,1,1,2,2,

%T 2,2,1,2,2,1,1,3,1,1,1,2,1,1,1,1,2,1,1,1,2,1,2,2,1,1,1,2,1,1,2,3,1,1,

%U 2,3,1,1,1,2,1,1,2,3,1,1,1,2,1,1,2,2,2

%N Number of modes in the prime factorization of n.

%C A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

%C a(n) depends only on the prime signature of n. - _Andrew Howroyd_, May 08 2023

%H Andrew Howroyd, <a href="/A362611/b362611.txt">Table of n, a(n) for n = 1..10000</a>

%F For n > 1, 1 <= a(n) << log n. - _Charles R Greathouse IV_, May 09 2023

%e The factorization of 450 is 2*3*3*5*5, modes {3,5}, so a(450) = 2.

%e The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so a(900) = 3.

%e The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so a(1500) = 1.

%e The factorization of 8820 is 2*2*3*3*5*7*7, modes {2,3,7}, so a(8820) = 3.

%t Table[x=Last/@If[n==1,0,FactorInteger[n]];Count[x,Max@@x],{n,100}]

%o (Python)

%o from sympy import factorint

%o def A362611(n): return list(v:=factorint(n).values()).count(max(v,default=0)) # _Chai Wah Wu_, May 08 2023

%o (PARI) a(n) = if(n==1, 0, my(f=factor(n)[,2], m=vecmax(f)); #select(v->v==m, f)) \\ _Andrew Howroyd_, May 08 2023

%Y Positions of first appearances are A002110.

%Y Positions of 1's are A356862, counted by A362608.

%Y Positions of terms > 1 are A362605, counted by A362607.

%Y For co-mode we have A362613, counted by A362615.

%Y This statistic (mode-count) has triangular form A362614.

%Y A027746 lists prime factors (with multiplicity).

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

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

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

%Y Cf. A000040, A000720, A002865, A215366, A327473, A327476, A359908.

%K nonn

%O 1,6

%A _Gus Wiseman_, May 05 2023