A067743 -id:A067743 - cp's OEIS frontend

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

Marc LeBrun, Jan 29 2002

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

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

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]

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.

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.

(* 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 *)

A067742(n) = {sumdiv(n, d, d2 = d^2; n / 2 < d2 && d2 <= n << 1)} \\ M. F. Hasler, May 12 2008

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

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

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

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

A302433 a(n) is the sum of the nonmiddle divisors of n.

Original entry on oeis.org

0, 2, 4, 5, 6, 7, 8, 13, 10, 18, 12, 21, 14, 24, 16, 27, 18, 36, 20, 33, 32, 36, 24, 50, 26, 42, 40, 45, 30, 61, 32, 59, 48, 54, 36, 85, 38, 60, 56, 77, 42, 83, 44, 84, 64, 72, 48, 110, 50, 88, 72, 98, 54, 105, 72, 105, 80, 90, 60, 152, 62, 96, 88, 119, 84, 127, 68, 126, 96, 127, 72, 172, 74, 114, 124

Offset: 1

Omar E. Pol, Apr 07 2018

nonn

The nonmiddle divisors of n are here the divisors of n that are not in the half-open interval [sqrt(n/2), sqrt(n*2)).

			For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12] and the middle divisors of 12 are [3, 4]. The divisors of 12 that are not middle divisors are called here the "nonmiddle" divisors of 12; they are [1, 2, 6, 12] and the sum of them is 1 + 2 + 6 + 12 = 21, so a(12) = 21.

Cf. A000203, A067742, A067743, A071090.

Table[DivisorSum[n, # &, Or[#^2 < n/2, #^2 >= 2 n] &], {n, 75}] (* Michael De Vlieger, Apr 21 2018 *)

a(n) = sumdiv( n, d, d*(d*d= 2*n)); \\ Michel Marcus, Apr 20 2018

a(n) = A000203(n) - A071090(n).

A375038 Irregular triangle read by rows T(n,k), n >= 2, k >= 1, in which row n lists the nonmiddle divisors of n.

Original entry on oeis.org

2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 2, 6, 12, 1, 13, 1, 2, 7, 14, 1, 15, 1, 2, 8, 16, 1, 17, 1, 2, 6, 9, 18, 1, 19, 1, 2, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 8, 12, 24, 1, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 14, 28

Offset: 2

Omar E. Pol, Jul 28 2024

Except the 1, all positive integers have nonmiddle divisors.

The nonmiddle divisors of n are here the divisors of n that are not in the half-open interval [sqrt(n/2), sqrt(n*2)).

			Triangle begins starting in row n = 2:
  2;
  1, 3;
  1, 4;
  1, 5;
  1, 6;
  1, 7;
  1, 4, 8;
  1, 9;
  1, 2, 5, 10;
  1, 11;
  1, 2, 6, 12;
  ...
For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12] and the middle divisors are [3, 4], so the nonmiddle divisors are [1, 2, 6, 12], the same as the row n = 12 of the triangle.

Amiram Eldar, Table of n, a(n) for n = 2..6375 (rows for n = 2..1000, flattened)

Nonzero terms of A375037.
The sum of row n is A302433(n).
The number of terms in row n is A067743(n).
Column 1 gives A054977.
Cf. A027750, A067742, A071090, A299761, A303297.

row[n_] := Select[Divisors[n], !(Sqrt[n/2] <= # < Sqrt[2*n]) &]; Table[row[n], {n, 2, 28}] // Flatten (* Amiram Eldar, Jul 29 2024 *)

A375037 Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n lists the divisors of n but every middle divisor is replaced with zero.

Original entry on oeis.org

0, 0, 2, 1, 3, 1, 0, 4, 1, 5, 1, 0, 0, 6, 1, 7, 1, 0, 4, 8, 1, 0, 9, 1, 2, 5, 10, 1, 11, 1, 2, 0, 0, 6, 12, 1, 13, 1, 2, 7, 14, 1, 0, 0, 15, 1, 2, 0, 8, 16, 1, 17, 1, 2, 0, 6, 9, 18, 1, 19, 1, 2, 0, 0, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 0, 0, 8, 12, 24

Offset: 1

Omar E. Pol, Jul 28 2024

The nonzero terms in row n are the nonmiddle divisors of n.

The nonmiddle divisors of n are here the divisors of n that are not in the half-open interval [sqrt(n/2), sqrt(n*2)).

			Triangle begins:
  0;
  0, 2;
  1, 3;
  1, 0, 4;
  1, 5;
  1, 0, 0, 6;
  1, 7;
  1, 0, 4, 8;
  1, 0, 9;
  1, 2, 5, 10;
  1, 11;
  1, 2, 0, 0, 6, 12;
  ...
For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12] and the middle divisors are [3, 4], but here the middle divisors are replaced with zeros, so the 12th row of the triangle is [1, 2, 0, 0, 6, 12].

Row sums give A302433.
Nonzero terms give A375038.
Row lengths give A000005.
The number of zeros in row n is A067742(n).
The number of nonzero terms in row n is A067743(n).
Cf. A000005, A027750, A067742, A299761, A303297.

row[n_] := Divisors[n] /. {x_?(Sqrt[n/2] <= # < Sqrt[2*n] &) -> 0}; Table[row[n], {n, 1, 24}] // Flatten (* Amiram Eldar, Jul 29 2024 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A302433 a(n) is the sum of the nonmiddle divisors of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A375038 Irregular triangle read by rows T(n,k), n >= 2, k >= 1, in which row n lists the nonmiddle divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A375037 Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n lists the divisors of n but every middle divisor is replaced with zero.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica