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A067742 Number of middle divisors of n, i.e., divisors in the half-open interval [sqrt(n/2), sqrt(n*2)). 27
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 (list; graph; refs; listen; history; text; internal format)
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 n-th 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 two-dimensional 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 number-theoretical 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 two-dimensional 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)^(k-1)*q^binomial(k+1, 2)/(1-q^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(n-1), 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[n-1]

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

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Last modified May 20 18:46 EDT 2018. Contains 304347 sequences. (Running on oeis4.)