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A365406
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Numbers j whose largest divisor <= sqrt(j) is a power of 2.
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6
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1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 82, 83, 86, 88, 89, 92, 94, 96, 97, 101, 103, 104, 106, 107, 109, 112, 113, 116, 118, 122, 124, 127, 128, 131, 134, 136, 137
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OFFSET
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1,2
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COMMENTS
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Also indices of the powers of 2 in A033676.
Also numbers in increasing order from the columns k of A163280 where k is a power of 2.
Observation: at least the first 82 terms of the subsequence of terms with no middle divisors (that is 3, 5, 7, 10, ...) coincide with at least the first 82 terms of A246955.
For the definition of middle divisor see A067742.
Most of the early terms are in A342081, which consists of powers of 2 together with products of a prime and a power of 2 where the prime is the larger. The exceptions are 24, 72, 80, 96, 112, ... .
The odd terms clearly consist of 1 and the odd primes. We can fully characterize the even terms by their A290110 values, which depend on the relative sizes of a number's divisors. A290110 provides a refinement of the classification of numbers by prime signature (cf. A212171): see the example below for numbers with the same prime signature as 48.
(End)
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LINKS
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EXAMPLE
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The table below looks at numbers j with prime signature (4, 1), showing the presence of j and its characterization by A290110(j):
48 = 2^4 * 3 16 no
80 = 2^4 * 5 21 yes
112 = 2^4 * 7 21 yes
162 = 2 * 3^4 36 no
176 = 2^4 * 11 38 no
208 = 2^4 * 13 38 no
272 = 2^4 * 17 51 yes
304 = 2^4 * 19 51 yes
368 = 2^4 * 23 51 yes
...
Clearly any odd composite number is exempted, for example:
891 = 3^4 * 11 21 no
6723 = 3^4 * 83 51 no
Note that A290110(j) = 36 for j = 2 * p^4, prime p; and A290110(j) = 51 for j = 2^4 * p, prime p >= 17.
(End)
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MATHEMATICA
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q[n_] := Module[{d = Divisors[n], mid}, mid = d[[Ceiling[Length[d]/2]]]; mid == 2^IntegerExponent[mid, 2]]; Select[Range[150], q] (* Amiram Eldar, Oct 11 2023 *)
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PROG
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(PARI) f(n) = local(d); if(n<2, 1, d=divisors(n); d[(length(d)+1)\2]); \\ A033676
isp2(n) = 2^logint(n, 2) == n;
(Python)
from itertools import count, islice
from sympy import divisors
def A365406_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda i:(a:=(d:=divisors(i))[len(d)-1>>1])==1<<a.bit_length()-1, count(max(startvalue, 1)))
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CROSSREFS
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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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