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A362613
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Number of co-modes in the prime factorization of n.
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44
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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, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 2, 2, 2
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OFFSET
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1,6
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COMMENTS
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First differs from A327500 at n = 180.
First differs from A351946 at n = 180.
First differs from A353507 at n = 180.
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}.
a(n) depends only on the prime signature of n. - Andrew Howroyd, May 08 2023
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LINKS
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EXAMPLE
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The factorization of 180 is 2*2*3*3*5, co-modes {5}, so a(180) = 1.
The factorization of 900 is 2*2*3*3*5*5, co-modes {2,3,5}, so a(900) = 3.
The factorization of 8820 is 2*2*3*3*5*7*7, co-modes {5}, so a(8820) = 1.
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MATHEMATICA
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Table[x=Last/@If[n==1, 0, FactorInteger[n]]; Count[x, Min@@x], {n, 100}]
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PROG
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(Python)
from sympy import factorint
v = factorint(n).values()
w = min(v, default=0)
return sum(1 for e in v if e<=w) # Chai Wah Wu, May 08 2023
(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
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CROSSREFS
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Positions of first appearances are A002110.
Counting partitions by this statistic (co-mode count) gives A362615.
A027746 lists prime factors (with multiplicity).
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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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