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%I A365406 #57 Oct 30 2023 11:06:51
%S A365406 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,
%T A365406 37,38,41,43,44,46,47,52,53,58,59,61,62,64,67,68,71,72,73,74,76,79,80,
%U A365406 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
%N A365406 Numbers j whose largest divisor <= sqrt(j) is a power of 2.
%C A365406 Also indices of the powers of 2 in A033676.
%C A365406 Also numbers in increasing order from the columns k of A163280 where k is a power of 2.
%C A365406 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.
%C A365406 For the definition of middle divisor see A067742.
%C A365406 From _Peter Munn_, Oct 26 2023: (Start)
%C A365406 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, ... .
%C A365406 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.
%C A365406 (End)
%H A365406 David A. Corneth, <a href="/A365406/b365406.txt">Table of n, a(n) for n = 1..10000</a>
%e A365406 From _Peter Munn_, Oct 26 2023: (Start)
%e A365406 The table below looks at numbers j with prime signature (4, 1), showing the presence of j and its characterization by A290110(j):
%e A365406     j             A290110(j)  present
%e A365406     48 = 2^4 * 3      16         no
%e A365406     80 = 2^4 * 5      21        yes
%e A365406    112 = 2^4 * 7      21        yes
%e A365406    162 = 2 * 3^4      36         no
%e A365406    176 = 2^4 * 11     38         no
%e A365406    208 = 2^4 * 13     38         no
%e A365406    272 = 2^4 * 17     51        yes
%e A365406    304 = 2^4 * 19     51        yes
%e A365406    368 = 2^4 * 23     51        yes
%e A365406   ...
%e A365406 Clearly any odd composite number is exempted, for example:
%e A365406    891 = 3^4 * 11     21         no
%e A365406   6723 = 3^4 * 83     51         no
%e A365406 Note that A290110(j) = 36 for j = 2 * p^4, prime p; and A290110(j) = 51 for j = 2^4 * p, prime p >= 17.
%e A365406 (End)
%t A365406 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 *)
%o A365406 (PARI) f(n) = local(d); if(n<2, 1, d=divisors(n); d[(length(d)+1)\2]); \\ A033676
%o A365406 isp2(n) = 2^logint(n,2) == n;
%o A365406 isok(k) = isp2(f(k)); \\ _Michel Marcus_, Oct 11 2023
%o A365406 (Python)
%o A365406 from itertools import count, islice
%o A365406 from sympy import divisors
%o A365406 def A365406_gen(startvalue=1): # generator of terms >= startvalue
%o A365406     return filter(lambda i:(a:=(d:=divisors(i))[len(d)-1>>1])==1<<a.bit_length()-1,count(max(startvalue,1)))
%o A365406 A365406_list = list(islice(A365406_gen(),30)) # _Chai Wah Wu_, Oct 18 2023
%Y A365406 Cf. A342081 (a subsequence), A365408 (complement), A365716 (characteristic function).
%Y A365406 Cf. A033676, A067742, A071561, A163280, A212171, A246955, A290110.
%K A365406 nonn
%O A365406 1,2
%A A365406 _Omar E. Pol_, Oct 10 2023

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