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

3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 111, 113, 114

Offset

1,1

Comments

Numbers k such that A071090(k) is 0.

Conjecture: lim_{n->oo} a(n)/n = 4/3.

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

Cf. A071090, A071562, A071563, A237048, A237270, A237271, A237593, A241008.

Sequence in context: A204827 A366882 A168252 * A241561 A241008 A335126

Adjacent sequences: A071558 A071559 A071560 * A071562 A071563 A071564

Keyword

nonn

Author

Robert G. Wilson v, May 30 2002

Status

approved