

A067742


Number of middle divisors of n, i.e., divisors in the halfopen interval [sqrt(n/2), sqrt(n*2)).


29



1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 2, 1, 0, 1, 0, 2, 0, 0, 0, 2, 1, 0, 0, 2, 0, 2, 0, 1, 0, 0, 2, 1, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 1, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 2, 0, 1, 2, 1, 0, 0, 0, 2, 0
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OFFSET

1,6


COMMENTS

From Omar E. Pol, Feb 06 2017: (Start)
Conjecture 1: a(n) is also the difference between the number of odd divisors of n less than sqrt(2*n) and the number of odd divisors of n greater than sqrt(2*n).
Conjecture 2: Also, alternating row sums of A237048.
Conjecture 3: Also central column of A249351.
Conjecture 4: The parity of this sequence is A053866 (also A093709, n>0), the same parity of A000203, A001227 and A000593.
Conjecture 5: Indices of odd terms gives A028982. Indices of even terms gives A028983.
Conjecture 6: a(n) is also the width of the terrace at the nth level in the main diagonal of the pyramid described in A245092.
Conjecture 7: a(n) is also the number of central subparts of the symmetric representation of sigma(n). For more information see A279387.
(End)
Conjecture 8: a(n) is also the difference between the number of partitions of n into an odd number of consecutive parts and the number of partitions of n into an even number of consecutive parts.  Omar E. Pol, Feb 24 2017


LINKS

R. Zumkeller, Table of n, a(n) for n = 1..10000
Robin Chapman, Kimmo Eriksson and Richard Stanley, On the Number of Divisors of n in a Special Interval: Problem 10847, Amer. Math. Monthly 109, (2002), p. 80.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a twodimensional torus, arXiv:1505.07229v3 [math.AG], 2015, see page 29 Remarks 6.8(b). [Note that a later version of this paper has a different title and different contents, and the numbertheoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a twodimensional torus, arXiv:1610.07793 [math.NT], 2016, see Remark 1.3.
J. E. Vatne, The sequence of middle divisors is unbounded, arXiv:1607.02122 [math.NT], 2016, shows that there is a subsequence diverging to infinity.


FORMULA

G.f.: Sum_{k>=1} (1)^(k1)*q^binomial(k+1, 2)/(1q^k).
a(A128605(n)) = n and a(m) <> n for m < A128605(n).  Reinhard Zumkeller, Mar 14 2007
It appears that a(n) = A240542(n)  A240542(n1), the difference between two adjacent Dyck paths as defined in A237593.  Hartmut F. W. Hoft, Feb 06 2017
Conjecture: a(n) = A082647(n)  A131576(n) = A001227(n)  2*A131576(n).  Omar E. Pol, Feb 06 2017


EXAMPLE

a(6)=2 because the 2 middle divisors of 6 (2 and 3) are between sqrt(3) and sqrt(12).


MATHEMATICA

(* number of middle divisors *)
a067742[n_] := Select[Divisors[n], Sqrt[n/2] <= # < Sqrt[2n] &]
a067742[115] (* data *)
(* Hartmut F. W. Hoft, Jul 17 2014 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, 1 &, n/2 <= #^2 < 2 n &]]; (* Michael Somos, Jun 04 2015 *)
(* support function a240542[] is defined in A240542 *)
b[n_] := a240542[n]  a240542[n1]
Map[b, Range[105]] (* data  Hartmut F. W. Hoft, Feb 06 2017 *)


PROG

(PARI) A067742(n) = {sumdiv(n, d, d2 = d^2; n / 2 < d2 && d2 <= n << 1)} \\ M. F. Hasler, May 12 2008
(PARI) a(n) = A067742(n) = {my(d = divisors(n), iu = il = #d \ 2); if(#d <= 2, return(n < 3)); while(d[il]^2 > n>>1, il); while(d[iu]^2 < (n<<1), iu++);
iu  il  1 + issquare(n/2)} \\ David A. Corneth, Apr 06 2018


CROSSREFS

Cf. A067743, A071090.
Cf. A071562 (lists all n such that a(n) is nonzero).
Relating to Dyck paths: A237593, A240542.
Sequence in context: A129561 A259895 A276479 * A302233 A214772 A242444
Adjacent sequences: A067739 A067740 A067741 * A067743 A067744 A067745


KEYWORD

easy,nonn


AUTHOR

Marc LeBrun, Jan 29 2002


STATUS

approved



