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Number of co-modes in the prime factorization of n.
44

%I #17 May 09 2023 08:55:17

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

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

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

%C First differs from A327500 at n = 180.

%C First differs from A351946 at n = 180.

%C First differs from A353507 at n = 180.

%C We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

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

%H Andrew Howroyd, <a href="/A362613/b362613.txt">Table of n, a(n) for n = 1..10000</a>

%e The factorization of 180 is 2*2*3*3*5, co-modes {5}, so a(180) = 1.

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

%e The factorization of 8820 is 2*2*3*3*5*7*7, co-modes {5}, so a(8820) = 1.

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

%o (Python)

%o from sympy import factorint

%o def A362613(n):

%o v = factorint(n).values()

%o w = min(v,default=0)

%o return sum(1 for e in v if e<=w) # _Chai Wah Wu_, May 08 2023

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

%Y Positions of first appearances are A002110.

%Y Positions of 1's are A359178, counted by A362610.

%Y Positions of terms > 1 are A362606, counted by A362609.

%Y For mode we have A362611, counted by A362614.

%Y Counting partitions by this statistic (co-mode count) gives A362615.

%Y A027746 lists prime factors (with multiplicity).

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

%Y Cf. A000040, A215366, A327473, A327476, A356862, A359908, A362605.

%K nonn

%O 1,6

%A _Gus Wiseman_, May 05 2023