A067742 Number of middle divisors of n, i.e., divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).

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

Offset: 1

Author

Marc LeBrun, Jan 29 2002

Keywords

• easy
• nonn

Comments

Comment from N. J. A. Sloane, Jan 03 2021: (Start)

Theorem 1: (i) a(n) = (number of odd divisors of n <= sqrt(2*n)) - (number of odd divisors of n > sqrt(2*n)).

(ii) Let r(n) = A003056(n). Then a(n) = (number of odd divisors of n <= r(n)) - (number of odd divisors of n > r(n)).

(iii) a(n) = Sum_{k=1..r(n)} (-1)^(k+1)*A237048(n,k).

(iv) a(n) is odd iff n is a square or twice a square (cf. A053866). Indices of odd terms give A028982. Indices of even terms give A028983.

The proofs are straightforward. These results were conjectured by Omar E. Pol in 2017. (End)

Theorem 2: a(n) is equal to 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. [Chapman et al., 2001; Hirschhorn and Hirschhorn, 2005]. - Omar E. Pol, Feb 06 2017

From Omar E. Pol, Feb 06 2017: (Start)

Conjecture 1: This is the central column of the isosceles triangle of A249351.

Conjecture 2: 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 3: a(n) is also the number of central subparts of the symmetric representation of sigma(n). For more information see A279387.

Conjectures 2 and 3 were proposed after Michel Marcus's conjecture in A237593. (End)

Conjectures 1, 2, and 3 are all true. - N. J. A. Sloane, Feb 11 2021

Examples

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

References

• Robin Chapman, Kimmo Eriksson and Richard Stanley, On the Number of Divisors of n in a Special Interval: Problem 10847, Amer. Math. Monthly 108:1 (Jan 2001), p. 77 (Proposal); 109:1 (Jan 2002), p. 80 (Solution). [Please do not delete this reference. - N. J. A. Sloane, Dec 16 2020]

Links

• Reinhard 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 108, (2001), p. 77; solution by Reiner Martin, Amer. Math. Monthly 109, (2002), p. 80.
• M. D. Hirschhorn and P. M. Hirschhorn, Partitions into Consecutive Parts, Mathematics Magazine 78.5 (2005): 396-396.
• 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.
• Omar E. Pol, Illustration of initial terms obtained geometrically
• 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.

Crossrefs

Cf. A001227, A003056, A028982, A028983, A053866, A067743, A071090 (sums of middle divisors), A082647, A128605, A131576.

Cf. also A071561 (positions of zeros), A071562 (positions of nonzeros), A299761 (middle divisors of n), A355143 (products of middle divisors).

Relation to Dyck paths: A237048, A237593, A240542 (partial sums), A245092, A249351, A279387, A348406.

Programs

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 *)

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

Python

from sympy import divisors
def A067742(n): return sum(1 for d in divisors(n,generator=True) if n <= 2*d**2 < 4*n) # Chai Wah Wu, Jun 09 2022

Formula

G.f.: Sum_{k>=1} (-1)^(k-1)*x^(k*(k+1)/2)/(1-x^k). (This g.f. corresponds to the assertion in Theorem 2.)

Another g.f., corresponding to the definition: Sum_{k>=1} x^(2*k*(k+1))*(1-x^(6*k^2))/(1-x^(2*k)) + Sum_{k>=0} x^((k+1)*(2*k+1))*(1-x^((2*k+1)*(3*k+2)))/(1-x^(2*k+1)). - N. J. A. Sloane, Jan 04 2021

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

The above conjecture is essentially the same as Michel Marcus's conjecture in A237593. - Omar E. Pol, Dec 20 2020

Conjecture: a(n) = A082647(n) - A131576(n) = A001227(n) - 2*A131576(n). - Omar E. Pol, Feb 06 2017

a(n) = A348406(n) - 1. - Omar E. Pol, Oct 29 2021

a(n) = A000005(n) - A067743(n). - Omar E. Pol, Jun 11 2022

Extensions

Edited by N. J. A. Sloane, Jan 03 2021