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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)). 25
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)

Conjectures:

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).

Also, alternating row sums of A237048.

Also central column of A249351.

The parity of this sequence is A053866 (also A093709, n>0), the same parity of A000203, A001227 and A000593.

Indices of odd terms gives A028982. Indices of even terms gives A028983.

a(n) is also the width of the terrace at the n-th level in the main diagonal of the pyramid described in A245092.

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

(End)

Conjecture: 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.

C. Kassel and C. Reutenauer, On the Zeta Functions of Punctual Hilbert schemes of a Two-Dimensional Torus, arXiv:1505.07229 [math.AG], 2015, see page 28 Remarks 6.9(b). [Note that version 1 mentions the OEIS 16 times. Later versions of this paper have a different title and different contents.]

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((-1)^(k-1)*q^(k+1 choose 2)/(1-q^k), k, 1, inf).

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 Dycks 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) A067743(n)=sumdiv( n, d, d*d<n/2 || d*d >= 2*n ) \\ M. F. Hasler, May 12 2008

CROSSREFS

Cf. A067743, A071090.

Cf. A071562 (lists all n such that a(n) is nonzero).

Cf. A237593, A240542 (relating to Dyck paths).

Sequence in context: A129561 A259895 A276479 * A214772 A242444 A236441

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 February 25 03:32 EST 2018. Contains 299630 sequences. (Running on oeis4.)