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A362611
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Number of modes in the prime factorization of n.
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53
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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, 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, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 2
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OFFSET
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1,6
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COMMENTS
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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}.
a(n) depends only on the prime signature of n. - Andrew Howroyd, May 08 2023
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LINKS
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FORMULA
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EXAMPLE
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The factorization of 450 is 2*3*3*5*5, modes {3,5}, so a(450) = 2.
The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so a(900) = 3.
The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so a(1500) = 1.
The factorization of 8820 is 2*2*3*3*5*7*7, modes {2,3,7}, so a(8820) = 3.
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MATHEMATICA
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Table[x=Last/@If[n==1, 0, FactorInteger[n]]; Count[x, Max@@x], {n, 100}]
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PROG
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(Python)
from sympy import factorint
def A362611(n): return list(v:=factorint(n).values()).count(max(v, default=0)) # Chai Wah Wu, May 08 2023
(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
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CROSSREFS
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Positions of first appearances are A002110.
This statistic (mode-count) has triangular form A362614.
A027746 lists prime factors (with multiplicity).
A362606 ranks partitions with more than one co-mode, counted by A362609.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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