A302433 -id:A302433 - cp's OEIS frontend

A071090 Sum of middle divisors of n.

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

Offset: 1

N. J. A. Sloane, May 27 2002

Divisors are in the half-open interval [sqrt(n/2), sqrt(n*2)).

Row sums of A299761. - Omar E. Pol, Jun 11 2022

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A000203, A067742, A071561, A071562, A299761, A302433.

A071090 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if d^2 >= n/2 and d^2 < n*2 then
            a := a+d ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jun 18 2015

Table[Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &], {n, 1, 95}]

a(n)=sumdiv(n,d, if(d^2>=n/2 && d^2<2*n, d, 0)) \\ Charles R Greathouse IV, Aug 01 2016

a(n) = A000203(n) - A302433(n). - Omar E. Pol, Jun 11 2022

A067743 Number of divisors of n not in the half-open interval [sqrt(n/2), sqrt(n*2)).

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 4, 2, 5, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 5, 4, 4, 2, 8, 2, 4, 4, 6, 2, 6, 2, 6, 4, 4, 2, 8, 2, 5, 4, 6, 2, 6, 4, 6, 4, 4, 2, 10, 2, 4, 4, 6, 4, 6, 2, 6, 4, 6, 2, 9, 2, 4, 6, 6, 2, 8, 2, 8, 4, 4, 2, 10, 4, 4, 4, 6, 2, 10, 2, 6, 4, 4, 4, 10, 2, 5, 4, 8, 2, 8

Offset: 1

Marc LeBrun, Jan 29 2002

From Max Alekseyev, May 13 2008: (Start)
Direct proof of Joerg Arndt's g.f. (see formula section).
We need to count divisors d|n such that d^2<=n/2 or d^2>2n. In the latter case, let's switch to co-divisor, replacing d with n/d.
Then we need to find the total count of: 1) divisors d|n such that 2d^2<=n; 2) divisors d|n such that 2d^2Let d|n and 2d^2<=n. Then n-2d^2 must be a multiple of d, i.e., n-2d^2=td for some integer t>=0.
Moreover it is easy to see that 1) is equivalent to n = 2d^2 + td for some integer t>=0. Therefore the answer for 1) is the coefficient of z^n in Sum_{d>=1} Sum_{t>=0} x^(2d^2 + td) = Sum_{d>=1} x^(2d^2)/(1 - x^d).
Similarly, the answer for 2) is Sum_{d>=1} x^(2d^2)/(1 - x^d) * x^d.
Therefore the g.f. for A067743 is Sum_{d>=1} x^(2d^2)/(1 - x^d) + Sum_{d>=1} x^(2d^2)/(1 - x^d) * x^d = Sum_{d>=1} x^(2d^2)/(1 - x^d) * (1 + x^d), as proposed. (End)
a(n) is odd if and only if n is in A001105. - Robert Israel, Oct 05 2020
Number of nonmiddle divisors of n. - Omar E. Pol, Jun 11 2022

			a(6)=2 because 2 divisors of 6 (i.e., 1 and 6) fall outside sqrt(3) to sqrt(12).

Robert Israel, Table of n, a(n) for n = 1..10000
Robin Chapman, Kimmo Ericksson, Richard P. Stanley and Reiner Martin, On the Number of Divisors of n in a Special Interval: Problem 10847, The American Mathematical Monthly, Vol. 109, No. 1 (Jan., 2002), p. 80.

Cf. A067742, A000005, A001105, A302433.

f:=proc(n) nops(select(t -> t^2 < n/2 or t^2 >= 2*n, numtheory:-divisors(n))) end proc:
map(f, [$1..200]); # Robert Israel, Oct 05 2020

hoi[n_]:=Length[DeleteCases[Divisors[n],?(Sqrt[n/2]<=#<Sqrt[2*n]&)]]; Array[ hoi,110] (* _Harvey P. Dale, Aug 22 2020 *)

A067743(n)=sumdiv( n,d, d*d= 2*n ) \\ M. F. Hasler, May 12 2008

a(n) = A000005(n) - A067742(n).

G.f.: Sum_{k>=1} z^(2*k^2)*(1+z^k)/(1-z^k). - Joerg Arndt, May 12 2008

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 *)

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A071090 Sum of middle divisors of n.

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A067743 Number of divisors of n not in the half-open interval [sqrt(n/2), sqrt(n*2)).

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

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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.

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