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A364061
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Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.
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9
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2, 4, 8, 16, 18, 32, 50, 54, 64, 98, 108, 128, 162, 242, 250, 256, 324, 338, 450, 486, 500, 512, 578, 648, 686, 722, 882, 972, 1024, 1058, 1250, 1350, 1372, 1458, 1682, 1922, 1944, 2048, 2178, 2250, 2450, 2500, 2646, 2662, 2738, 2916, 3042, 3362, 3698, 3888
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OFFSET
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1,1
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COMMENTS
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Also numbers whose multiset of prime factors has unique co-mode 2. Here, a co-mode in a multiset is an element that appears at most as many times as each of the other elements. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.
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LINKS
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EXAMPLE
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The terms together with their prime factors begin:
2 = 2
4 = 2*2
8 = 2*2*2
16 = 2*2*2*2
18 = 2*3*3
32 = 2*2*2*2*2
50 = 2*5*5
54 = 2*3*3*3
64 = 2*2*2*2*2*2
98 = 2*7*7
108 = 2*2*3*3*3
128 = 2*2*2*2*2*2*2
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MAPLE
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filter:= proc(n) local F, F2, Fo;
F:= ifactors(n)[2];
F2, Fo:= selectremove(t -> t[1]=2, F);
Fo = [] or F2[1, 2] < min(Fo[.., 2])
end proc:
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MATHEMATICA
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prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
comodes[ms_]:=Select[Union[ms], Count[ms, #]<=Min@@Length/@Split[ms]&];
Select[Range[100], comodes[prifacs[#]]=={2}&]
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PROG
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(Python)
from sympy import factorint
from itertools import count, islice
def A364061_gen(startvalue=2): # generator of terms >= startvalue
return filter(lambda n:(l:=(~n&n-1).bit_length()) < min(factorint(m:=n>>l).values(), default=0) or m==1, count(max(startvalue+startvalue&1, 2), 2))
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CROSSREFS
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Partitions of this type are counted by A364062.
Cf. A000265, A007814, A327473, A327476, A362616, A360014, A363722, A363723, A363725, A363727, A363730.
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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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