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A071561 Numbers with no middle divisors (cf. A071090).

%I A071561 #71 Jul 27 2024 23:17:59
%S A071561 3,5,7,10,11,13,14,17,19,21,22,23,26,27,29,31,33,34,37,38,39,41,43,44,
%T A071561 46,47,51,52,53,55,57,58,59,61,62,65,67,68,69,71,73,74,75,76,78,79,82,
%U A071561 83,85,86,87,89,92,93,94,95,97,101,102,103,105,106,107,109,111,113,114
%N A071561 Numbers with no middle divisors (cf. A071090).
%C A071561 Numbers k such that A071090(k) is 0.
%C A071561 Conjecture: lim_{n->oo} a(n)/n = 4/3.
%C A071561 Regarding the above conjecture, numerical calculations suggest that this limit is smaller than 4/3. See A071540. - _Amiram Eldar_, Jul 27 2024
%C A071561 Also numbers n with the property that the number of parts in the symmetric representation of sigma(n) is even. - _Michel Marcus_ and _Omar E. Pol_, Apr 25 2014 [For a proof see the link. - _Hartmut F. W. Hoft_, Sep 09 2015]
%C A071561 Middle divisors are divisors d with sqrt(k/2) <= d < sqrt(2k). - _Michael B. Porter_, Oct 19 2018
%H A071561 Iain Fox, <a href="/A071561/b071561.txt">Table of n, a(n) for n = 1..10000</a>
%H A071561 Hartmut F. W. Hoft, <a href="/A071561/a071561.pdf">On the symmetric spectrum of odd divisors of a number</a>, (2015), (Note that in this paper, A241561 should be replaced with A071561, and A241562 should be replaced with A071562. Also note that "the symmetric spectrum of odd divisors of a number" seems to be an attempt to call with a new name to a diagram known since 2014 as "the symmetric representation of sigma(n)"). - _Omar E. Pol_, Oct 08 2018
%H A071561 José Manuel Rodríguez Caballero, <a href="https://arxiv.org/abs/1709.09617">Elementary number-theoretical statements proved by Language Theory</a>, arXiv:1709.09617 [math.LO], 2017.
%H A071561 J. M. Rodríguez Caballero, <a href="https://dx.doi.org/10.1007/978-3-319-66396-8_23">Symmetric Dyck Paths and Hooley's Δ-Function</a>, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.
%e A071561 From _Michael B. Porter_, Oct 19 2018: (Start)
%e A071561 The divisors of 21 are 1, 3, 7, and 21.  Since none of these are between sqrt(21/2) = 3.24... and sqrt(2*21) = 6.48..., 21 is in the sequence.
%e A071561 The divisors of 20 are 1, 2, 4, 5, 10, and 20.  Since 4 and 5 are both between sqrt(20/2) = 3.16... and sqrt(2*20) = 6.32..., 20 is not in the sequence. (End)
%t A071561 f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[125], f[ # ] == 0 &]
%t A071561 (* Related to the symmetric representation of sigma *)
%t A071561 (* subsequence of even parts of number k for m <= k <= n *)
%t A071561 (* Function a237270[] is defined in A237270 *)
%t A071561 (* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *)
%t A071561 a071561[m_, n_]:=Select[Range[m, n], EvenQ[Length[a237270[#]]]&]
%t A071561 a071561[1, 114] (* data *)
%t A071561 (* _Hartmut F. W. Hoft_, Jul 07 2014 *)
%t A071561 Select[Range@ 120, Function[n, Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] == {}]] (* _Michael De Vlieger_, Jan 03 2017 *)
%o A071561 (PARI) is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1 \\ _Iain Fox_, Dec 19 2017
%o A071561 (PARI) is(n,f=factor(n))=my(t=(n+1)\2); fordiv(f,d, if(d^2>=t, return(d^2>2*n))); 0 \\ _Charles R Greathouse IV_, Jan 22 2018
%o A071561 (PARI) list(lim)=my(v=List(),t); forfactored(n=3,lim\1, t=(n[1]+1)\2; fordiv(n[2],d, if(d^2>=t, if(d^2>2*n[1], listput(v,n[1])); break))); Vec(v) \\ _Charles R Greathouse IV_, Jan 22 2018
%Y A071561 Cf. A071090, A071540, A071562, A071563, A237048, A237270, A237271, A237593, A241008.
%K A071561 nonn
%O A071561 1,1
%A A071561 _Robert G. Wilson v_, May 30 2002