OFFSET
1,1
COMMENTS
Numbers k such that A071090(k) is 0.
Conjecture: lim_{n->oo} a(n)/n = 4/3.
Regarding the above conjecture, numerical calculations suggest that this limit is smaller than 4/3. See A071540. - Amiram Eldar, Jul 27 2024
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]
Middle divisors are divisors d with sqrt(k/2) <= d < sqrt(2k). - Michael B. Porter, Oct 19 2018
LINKS
Iain Fox, Table of n, a(n) for n = 1..10000
Hartmut F. W. Hoft, On the symmetric spectrum of odd divisors of a number, (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
José Manuel Rodríguez Caballero, Elementary number-theoretical statements proved by Language Theory, arXiv:1709.09617 [math.LO], 2017.
J. M. Rodríguez Caballero, Symmetric Dyck Paths and Hooley's Δ-Function, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.
EXAMPLE
From Michael B. Porter, Oct 19 2018: (Start)
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.
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)
MATHEMATICA
f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[125], f[ # ] == 0 &]
(* Related to the symmetric representation of sigma *)
(* subsequence of even parts of number k for m <= k <= n *)
(* Function a237270[] is defined in A237270 *)
(* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *)
a071561[m_, n_]:=Select[Range[m, n], EvenQ[Length[a237270[#]]]&]
a071561[1, 114] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)
Select[Range@ 120, Function[n, Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] == {}]] (* Michael De Vlieger, Jan 03 2017 *)
PROG
(PARI) is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1 \\ Iain Fox, Dec 19 2017
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, May 30 2002
STATUS
approved