A280849 -id:A280849 - cp's OEIS frontend

A320137 Numbers that have only one middle divisor.

1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 49, 50, 64, 81, 98, 100, 121, 128, 162, 169, 196, 200, 225, 242, 256, 289, 324, 338, 361, 392, 441, 484, 512, 529, 578, 625, 676, 722, 729, 784, 841, 882, 961, 968, 1024, 1058, 1089, 1156, 1250, 1352, 1369, 1444, 1458, 1521, 1681, 1682, 1849, 1922, 1936, 2025, 2048, 2116

Offset: 1

Omar E. Pol, Oct 06 2018

nonn

Conjecture 1: sequence consists of numbers k with the property that the difference between the number of partitions of k into an odd number of consecutive parts and the number of partitions of k into an even number of consecutive parts is equal to 1.
Conjecture 2: sequence consists of numbers k with the property that the symmetric representation of sigma(k) has width 1 on the main diagonal.
Conjecture 3: all powers of 2 are in the sequence.
From Hartmut F. W. Hoft, May 24 2022: (Start)
Every number in this sequence is a square or twice a square, i.e., this sequence is a subsequence of A028982, and conjectures 2 and 3 are true (see the link for proofs). Furthermore, all odd numbers in this sequence are squares and form subsequences of A016754 and of A319529.
Every number k in this sequence has the form k = 2^m * q^2, m >= 0, q >= 1 odd, where for any divisor e of q^2 smaller than the largest divisor of q^2 that is less than or equal to row(q^2) = floor((sqrt(8*q^2 + 1) - 1)/2) the inequalities 2^(m+1) * e < row(n) hold (see the link for a proof).
The smallest odd square not in this sequence is 1225 = 35^2 = (5*7)^2 since it has the 3 middle divisors 25, 35, 49 and the width of the symmetric representation of sigma(1225) at the diagonal equals 3. However, the squares of odd primes in this sequence are a subsequence of A259417.
The smallest even square not in this sequence is 144 = 12^2 = (2*2*3)^2 since it has the 3 middle divisors 9, 12, 16 and the width of the symmetric representation of sigma(144) at the diagonal equals 3.
The smallest twice square not in this sequence is 72 = 2 * (2*3)^2 = 2^3 * 3^2 since it has the 3 middle divisors 6, 8, 9 and the width of the symmetric representation of sigma(72) at the diagonal equals 3.
Apart from the powers of 2 in the infinite first row, the numbers in the sequence can be arranged as an irregular triangle with each row containing the finitely many numbers q^2, 2 * q^2, 4 * q^2, ..., 2^m * q^2 satisfying the condition stated above, as shown below:
    1     2     4     8     16     32     64     128     256 ...
    9    18    36
   25    50   100   200
   49    98   196   392    784
   81   162   324
  121   242   484   968   1936   3872
  169   338   676  1352   2704   5408  10816
  225
  289   578  1156  2312   4624   9248  18496   36992
  361   722  1444  2888   5776  11552  23104   46208
  441   882
  529  1058  2116  4232   8464  16928  33856   67712  135424
  625  1250  2500  5000
  729  1458  2916
  841  1682  3364  6728  13456  26912  53824  107648  215296
  ...
(End)

			9 is in the sequence because 9 has only one middle divisor: 3.
On the other hand, in accordance with the first conjecture, 9 is in the sequence because there are two partitions of 9 into an odd number of consecutive parts: [9], [4, 3, 2], and there is only one partition of 9 into an even number of consecutive parts: [5, 4], therefore the difference of the number of those partitions is 2 - 1 = 1.
On the other hand, in accordance with the second conjecture, 9 is in the sequence because the symmetric representation of sigma(9) = 13 has width 1 on the main diagonal, as shown below in the first quadrant:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.                    | |
.                    | |
.                    | |
.                    |_|
.

Hartmut F. W. Hoft, Proofs of conjectures 2 and 3

Column 1 of A320051.
First differs from A028982 at a(14).
For the definition of middle divisors see A067742.
Cf. A000079, A071561, A071562, A237048, A237593, A240542, A245092, A249351 (widths), A279286, A279387, A280849, A281007, A299761, A299777, A303297, A319529, A319796, A319801, A319802, A320142.
Cf. A016754, A028982, A249223, A259417.

(* computation based on counts of divisors *)
middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#A237048 and A249223 for width at diagonal *)
a249223[n_] := Drop[FoldList[Plus, 0, Map[(-1)^(#+1) a237048[n, #]&, Range[Floor[(Sqrt[8n+1]-1)/2]]]], 1]
a320137W[n_] := Select[Range[n], Last[a249223[#]]==1&]
a320137W[2116]
(* Hartmut F. W. Hoft, May 24 2022 *)

A320142 Numbers that have exactly two middle divisors.

Original entry on oeis.org

6, 12, 15, 20, 24, 28, 30, 35, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 77, 80, 84, 88, 90, 91, 96, 99, 104, 108, 110, 112, 117, 126, 130, 132, 135, 140, 143, 150, 153, 154, 156, 160, 165, 168, 170, 176, 182, 187, 190, 192, 195, 198, 204, 208, 209, 210, 216, 220, 221, 224, 228, 231, 234, 238, 247, 255, 260

Offset: 1

Omar E. Pol, Oct 06 2018

nonn

Conjecture 1: numbers k with the property that the difference between the number of partitions of k into an odd number of consecutive parts and the number of partitions of k into an even number of consecutive parts is equal to 2.
Conjecture 2: numbers k with the property that symmetric representation of sigma(k) has width 2 on the main diagonal.
By the theorem in A067742 conjecture 2 is true. - Hartmut F. W. Hoft, Aug 18 2024

			15 is in the sequence because 15 has two middle divisors: 3 and 5.
On the other hand, in accordance with the first conjecture, 15 is in the sequence because there are three partitions of 15 into an odd number of consecutive parts: [15], [8, 7], [5, 4, 3, 2, 1], and there is only one partition of 15 into an even number of consecutive parts: [8, 7], therefore the difference of the number of those partitions is 3 - 1 = 2.
On the other hand, in accordance with the second conjecture, 15 is in the sequence because the symmetric representation of sigma(15) = 24 has width 2 on the main diagonal, as shown below in the fourth quadrant:
.                                _
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                          _ _ _|_|
.                      _ _| |      8
.                     |    _|
.                    _|  _|
.                   |_ _|  8
.                   |
.    _ _ _ _ _ _ _ _|
.   |_ _ _ _ _ _ _ _|
.                    8
.

Column 2 of A320051.
First differs from A001284 at a(19).
For the definition of middle divisors see A067742.
Cf. A071561, A071562, A237048, A237593, A240542, A245092, A249351 (widths), A279286, A279387, A280849, A281007, A299761, A299777, A303297, A319529, A319796, A319801, A319802, A320137.

a320142Q[k_] := Length[Select[Divisors[k], k/2<=#^2<2k&]]==2
a320142[n_] := Select[Range[n], a320142Q]
a320142[260] (* Hartmut F. W. Hoft, Aug 20 2024 *)

A281009 Number of odd divisors of n minus the number of middle divisors of n.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 4, 2, 2, 0, 2, 2, 4, 0, 2, 2, 2, 0, 4, 2, 2, 2, 2, 2, 4, 0, 2, 2, 2, 2, 4, 2, 2, 0, 2, 2, 4, 2, 2, 2, 4, 0, 4, 2, 2, 2, 2, 2, 4, 0, 4, 2, 2, 2, 4, 2, 2, 0, 2, 2, 6, 2, 2, 4, 2, 0, 4, 2, 2, 2, 4, 2, 4, 0, 2, 4, 2, 2, 4, 2, 4, 0, 2, 2, 4, 2, 2, 4, 2, 0, 8

Offset: 1

Omar E. Pol, Feb 20 2017

nonn

Conjecture 1: a(n) is also twice the number of odd divisors of n greater than sqrt(2*n).
Conjecture 2: a(n) is also the number of odd divisors of n less than sqrt(2*n) that are not middle divisors of n, plus the number of odd divisors of n greater than sqrt(2*n).
Conjecture 3: a(n) is also the total number of equidistant subparts in the symmetric representation of sigma(n).
The "equidistant subparts" are the subparts that are not the "central subparts".
For more information of the "subparts" see A279387.

			For n = 45 the divisors of 45 are [1, 3, 5, 9, 15, 45]. There are 6 odd divisors, and two of them [5 and 9] are also the middle divisors of 45, so a(45) = 6 - 2 = 4.
Other examples (conjectured):
2) There are two odd divisors of 45 that are greater than the square root of 2*45 = 9.4..., so a(45) = 2*2 = 4.
3) The 45th row of A237593 is [23, 8, 5, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 5, 8, 23], and the 44th row of the same triangle is [23, 8, 4, 3, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 8, 23], therefore between both symmetric Dyck paths (described in A237593 and A279387) there are two central subparts [27 and 1] and two pairs of equidistant subparts ([23, 23] and [2, 2]). The total number of equidistant subparts is equal to 4, so a(45) = 4. (the diagram of the symmetric representation of sigma(45) is too large to include).
4) The 45th row of A196020 is [89, 43, 27, 0, 13, 9, 0, 0, 1], hence the 45th row of A280850 is [23, 23, 27, 0, 2, 2, 0, 0, 1]. There are two central subparts [27 and 1] and two pairs of equidistant subparts ([23, 23] and [2, 2]). The total number of equidistant subparts is equal to 4, so a(45) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001227, A067742, A082647, A131576, A196020, A236104, A237048, A237593, A245092, A249351, A261699, A262626, A279667, A280849, A280850, A280940, A281005, A281007, A281008.

N:= 200: # to get a(1)..a(N)
A:= Vector(N):
for m from 1 to N by 2 do
  R:= [seq(k*m,k=1..N/m)];
  A[R]:= A[R] + Vector(nops(R),1);
od:
for m from 1 to N do
  R:= [seq(k*m, k= floor(m/2)+1..min(2*m,N/m))];
  A[R]:= A[R] - Vector(nops(R),1);
od:
convert(A,list); # Robert Israel, Feb 20 2017

Table[Count[#, d_ /; OddQ@ d] - Count[#, d_ /; Sqrt[n/2] <= d < Sqrt[2 n]] &@ Divisors@ n, {n, 120}] (* Michael De Vlieger, Feb 20 2017 *)

a(n) = A001227(n) - A067742(n).

Conjecture: a(n) = 2*A131576(n).

A281008 Least positive integer k with exactly n odd divisors greater than sqrt(2*k).

Original entry on oeis.org

1, 3, 21, 75, 105, 315, 495, 945, 1575, 2835, 3465, 4095, 11025, 17955, 10395, 23205, 17325, 24255, 31185, 36855, 51975, 61425, 45045, 108675, 143325, 121275, 184275, 155925, 135135, 176715, 239085, 315315, 294525, 225225, 606375, 626535, 405405, 700245, 1531530, 1351350, 2072070, 1289925, 855855

Offset: 0

Omar E. Pol, Feb 16 2017

nonn

Conjecture: a(n) is also the smallest number k having n pairs of equidistant subparts in the symmetric representation of sigma(k).
For more information about the "subparts" see A279387.
Observations about the known terms:
Observation 1: terms a(1)-a(51) are divisible by 3.
Observation 2: terms a(3)-a(51) are divisible by 5.

			a(3) = 75 because the divisors of 75 are [1, 3, 5, 15, 25, 75], and 75 has three odd divisors greater than the square root of 2*75 = 12.2..., and it is the smallest number with that property.
Other examples (conjectured):
2) The 75th row of A237593 is [38, 13, 7, 4, 3, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 3, 4, 7, 13, 38], and the 74th row of the same triangle is [38, 13, 6, 5, 3, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 3, 5, 6, 13, 38], therefore between both symmetric Dyck paths (described in A237593 and A279387) there are three pairs of equidistant subparts: [38, 38], [21, 21] and [3, 3]. That is the first row with that property, so a(3) = 75. (The diagram of the symmetric representation of sigma(75) is too large to include).
3) The 75th row of A196020 is [149, 73, 47, 0, 25, 19, 0, 0, 0, 5, 0], hence the 75th row of A280850 is [38, 38, 21, 0, 3, 3, 0, 0, 0, 21, 0]. There are three pairs of equidistant subparts [38, 38], [21, 21] and [3, 3]. That is the first row with that property, so a(3) = 75.
4) The 75th row of A237048 is [1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0]. The sum of the even-indexed terms is equal to 3. That is the first row with that property, so a(3) = 75.
5) The 75th row of A261699 is [1, 75, 3, 0, 5, 25, 0, 0, 0, 15, 0]. There are three even-indexed terms that are positive integers: [75, 25, 15]. That is the first row with that property, so a(3) = 75.

Charles R Greathouse IV, Table of n, a(n) for n = 0..200

Row 1 of A280849.

Cf. A000203, A001227, A008585, A067742, A082262, A131576, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A261699, A262626, A279387, A280850, A280940, A281005.

cnt[k_] := cnt[k] = DivisorSum[k, Boole[OddQ[#] && #>Sqrt[2k]]&]; a[n_] := a[n] = For[k = 1, True, k++, If[cnt[k]==n, Return[k]]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 16 2017 *)

a(n,{s=0},{q=1},{k=2},{w=1})={if(n<1,return(1));my(z,ii,F,d,L:list,V,p,ans:list);ans=List();if(q<1,q=1);if(k<2,k=2);while(k++,p=sqrt(2*k);F=factor(k);ii=vecsum(F[1,]);F=F[,1]~;L=List([1]);for(i=1,ii,forvec(y=vector(i,t,[1,#F]),d=prod(u=1,#y,F[y[u]]);if((d<=k)&&!(k%d),listput(L,d)),1));V=Set(Vec(L));if(n==sum(u=1,#V,(V[u]>p)&&(V[u]%2==!!w)),if(s,print1(V","));listput(ans,k);if(z++==q,if(#ans==1,return(k),return(Vec(ans))),n++)))} \\ with n>=1, "s" set to 1 also prints the divisors (of "w" version: 1 odd, 0 even) for the first "q" terms from the n-th, resuming their search with k>=2. - R. J. Cano, Feb 20 2017

a(n)=my(k,s); while(k++, s=sqrtint(2*k); if(sumdiv(k>>valuation(k,2), d, d>s)==n, return(k))) \\ Charles R Greathouse IV, Feb 20 2017

a(10)-a(30) from Jean-François Alcover, Feb 16 2017

a(31)-a(43) from Michael De Vlieger, Feb 18 2017

A320051 Square array read by antidiagonals upwards: T(n,k) is the n-th positive integer with exactly k middle divisors, n >= 1, k >= 0.

Original entry on oeis.org

3, 5, 1, 7, 2, 6, 10, 4, 12, 72, 11, 8, 15, 144, 120, 13, 9, 20, 288, 180, 1800, 14, 16, 24, 400, 240, 3528, 840, 17, 18, 28, 450, 252, 4050, 1080, 3600, 19, 25, 30, 576, 336, 5184, 1260, 7200, 2520, 21, 32, 35, 648, 360, 7056, 1440, 14112, 5040, 28800, 22, 36, 40, 800, 378, 8100, 1680, 14400, 5544

Offset: 1

Omar E. Pol, Oct 04 2018

This is a permutation of the natural numbers.
For the definition of middle divisors see A067742.
Conjecture 1: T(n,k) is also the n-th positive integer j with the property that the difference between the number of partitions of j into an odd number of consecutive parts and the number of partitions of j into an even number of consecutive parts is equal to k.
Conjecture 2: T(n,k) is also the n-th positive integer j with the property that the symmetric representation of sigma(j) has width k on the main diagonal.

			The corner of the square array begins:
   3,  1,  6,  72, 120, 1800,  840,  3600, 2520, 28800, ...
   5,  2, 12, 144, 180, 3528, 1080,  7200, 5040, ...
   7,  4, 15, 288, 240, 4050, 1260, 14112, ...
  10,  8, 20, 400, 252, 5184, 1440, ...
  11,  9, 24, 450, 336, 7056, ...
  13, 16, 28, 576, 360, ...
  14, 18, 30, 648, ...
  17, 25, 35, ...
  19, 32, ...
  21, ...
  ...
In accordance with the conjecture 1, T(1,0) = 3 because there is only one partition of 3 into an odd number of consecutive parts: [3], and there is only one partition of 3 into an even number of consecutive parts: [2, 1], therefore the difference of the number of those partitions is 1 - 1 = 0.
On the other hand, in accordance with the conjecture 2: T(1,0) = 3 because the symmetric representation of sigma(3) = 4 has width 0 on the main diagonal, as shown below:
.    _ _
.   |_ _|_
.       | |
.       |_|
.
In accordance with the conjecture 1, T(1,2) = 6 because there are three partitions of 6 into an odd number of consecutive parts: [6], [3, 2, 1], and there are no partitions of 6 into an even number of consecutive parts, therefore the difference of the number of those partitions is 2 - 0 = 2.
On the other hand, in accordance with the conjecture 2: T(1,2) = 6 because the symmetric representation of sigma(6) = 12 has width 2 on the main diagonal, as shown below:
.    _ _ _ _
.   |_ _ _  |_
.         |   |_
.         |_ _  |
.             | |
.             | |
.             |_|
.

Index entries for sequences that are permutations of the natural numbers

Row 1 is A128605.
Column 0 is A071561.
The union of the rest of the columns gives A071562.
Column 1 is A320137.
Column 2 is A320142.
For more information about the diagrams see A237593.
For tables of partitions into consecutive parts see A286000 and A286001.
Cf. A067742, A240542, A245092, A249351 (widths), A262626, A279286, A280849, A281007, A299761, A299777, A303297, A319529, A319796, A319801, A319802.

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A320137 Numbers that have only one middle divisor.

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A320142 Numbers that have exactly two middle divisors.

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A281009 Number of odd divisors of n minus the number of middle divisors of n.

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A281008 Least positive integer k with exactly n odd divisors greater than sqrt(2*k).

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A320051 Square array read by antidiagonals upwards: T(n,k) is the n-th positive integer with exactly k middle divisors, n >= 1, k >= 0.

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