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A360015
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Numbers whose exponent of 2 in their canonical prime factorization is equal to the maximal exponent.
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24
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1, 2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 52, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 92, 94, 96, 100, 102, 104, 106, 110, 112, 114, 116, 118, 120, 122, 124, 128, 130, 132, 134, 136, 138
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OFFSET
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1,2
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COMMENTS
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The powers of 2 (A000079) are all terms.
The product of any two terms (not necessarily distinct) is also a term.
This sequence is a disjoint union of {1} and the subsequences of numbers m of the form 2^(k-1)*o where o = A000265(m), the odd part of m, is a k-free number, for k >= 2. These subsequences include, for k = 2, numbers of the form 2*o where o is an odd squarefree number (A056911); for k = 3, numbers of the form 4*o where o is an odd cubefree number; etc.
The asymptotic density of this sequence is Sum_{k>=2} 1/(zeta(k)*(2^k-1)) = 0.44541445377638761933... .
The asymptotic mean of the exponent of 2 in the prime factorization of the terms of this sequence is Sum_{k>=2} (k-1)/(zeta(k)*(2^k-1)) = 0.93691473348959419722... .
Also numbers whose multiset of prime factors has low (i.e. least) mode 2. Here, a mode in a multiset is an element that appears at least as many times as each of the others; for example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}. - Gus Wiseman, Jul 14 2023
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LINKS
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FORMULA
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EXAMPLE
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108 = 2*2*3*3*3 is missing because its mode is not 2.
180 = 2*2*3*3*5 is present because it has low mode 2.
The terms together with their prime factorizations begin:
1 =
2 = 2
4 = 2*2
6 = 2*3
8 = 2*2*2
10 = 2*5
12 = 2*2*3
14 = 2*7
16 = 2*2*2*2
20 = 2*2*5
22 = 2*11
24 = 2*2*2*3
26 = 2*13
28 = 2*2*7
30 = 2*3*5
32 = 2*2*2*2*2
34 = 2*17
36 = 2*2*3*3
(End)
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MATHEMATICA
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q[n_] := IntegerExponent[n, 2] == Max[FactorInteger[n][[;; , 2]]]; q[1] = True; Select[Range[150], q]
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PROG
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(PARI) is(n) = n == 1 || vecmax(factor(n)[, 2]) == valuation(n, 2);
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CROSSREFS
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Partitions of this type are counted by A241131.
For unique minimal prime exponent we have A364061, counted by A364062.
A027746 lists prime factors (with multiplicity).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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