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A303297 List of middle divisors: for every positive integer that has middle divisors, add its middle divisors to the sequence.

%I A303297 #50 Aug 27 2023 23:24:37
%S A303297 1,1,2,2,3,2,3,3,4,3,5,4,3,4,5,4,6,5,4,7,5,6,4,5,7,6,5,8,6,7,5,9,6,8,
%T A303297 7,5,6,9,7,8,6,10,7,9,8,6,11,7,10,6,8,9,7,11,8,10,9,7,12,8,11,9,10,7,
%U A303297 13,8,12,7,9,11,10,8,13,9,12,10,11,8,14,9,13,8,10,12,15,11,9,14,8,10,13,11,12,9,15
%N A303297 List of middle divisors: for every positive integer that has middle divisors, add its middle divisors to the sequence.
%C A303297 The middle divisors of k (see A299761) are the divisors in the half-open interval [sqrt(k/2), sqrt(k*2)), k >= 1.
%H A303297 Michel Marcus, <a href="/A303297/b303297.txt">Table of n, a(n) for n = 1..6934</a>
%H A303297 Michael De Vlieger, <a href="/A303297/a303297.png">Plot (n,d) at (x,y)</a> for middle divisors d | n and n <= 2^16.
%H A303297 Michael De Vlieger, <a href="/A303297/a303297_1.png">Plot (n,d) at (x,y)</a> for middle divisors d | n and n <= 345, labeling n, and showing composite prime powers in gold, squarefree composites in green, numbers neither squarefree nor composite in blue, and highlighting products of composite prime powers in large light blue.
%H A303297 Michael De Vlieger, <a href="/A303297/a303297_2.png">Plot (n,d) at (x,y)</a> for middle divisors d | n and n <= 2^16, with same color function as above so as to show patterns according to prime power decomposition of n.
%e A303297 The middle divisor of 1 is 1, so a(1) = 1.
%e A303297 The middle divisor of 2 is 1, so a(2) = 1.
%e A303297 There are no middle divisors of 3.
%e A303297 The middle divisor of 4 is 2, so a(3) = 2.
%e A303297 There are no middle divisors of 5.
%e A303297 The middle divisors of 6 are 2 and 3, so a(4) = 2 and a(5) = 3.
%e A303297 There are no middle divisors of 7.
%e A303297 The middle divisor of 8 is 2, so a(6) = 2.
%e A303297 The middle divisor of 9 is 3, so a(7) = 3.
%e A303297 There are no middle divisors of 10.
%e A303297 There are no middle divisors of 11.
%e A303297 The middle divisors of 12 are 3 and 4, so a(8) = 3 and a(9) = 4.
%t A303297 Table[Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] /. {} -> Nothing, {n, 135}] // Flatten (* _Michael De Vlieger_, Jun 14 2018 *)
%o A303297 (PARI) lista(nn) = {my(list = List()); for (n=1, nn, my(v = select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2))), divisors(n))); for (i=1, #v, listput(list, v[i]));); Vec(list);} \\ _Michel Marcus_, Mar 26 2023
%Y A303297 Concatenate the nonzero rows of A299761 (that is, the nonzero terms of A299761).
%Y A303297 Cf. A027750, A067742, A071090, A071562, A281007, A299777.
%K A303297 nonn,look
%O A303297 1,3
%A A303297 _Omar E. Pol_, Apr 30 2018