A128605 -id:A128605 - 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

A282197 a(n) is the smallest number d if the point (d,d) is shared by exactly n different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.

Original entry on oeis.org

1, 2, 7, 15, 52, 102, 296, 371, 455, 929, 1853, 2034, 4517, 4797, 5829, 6146, 6948, 17577, 19818, 18915, 60349, 78369, 113010, 110185, 91650, 85171, 311321, 123788, 823049, 128596, 1650408, 1136865, 415355, 906771, 2897535

Offset: 1

Hartmut F. W. Hoft, Feb 08 2017

This sequence is not monotone since a(19) = 19818 > 18915 = a(20).
Additional values smaller than 5000000 are a(37) = 1751785, a(38) = 1786732, a(39) = 1645139, a(41) = 1308771 and a(44) = 3329668.
Sequence A128605 of first occurrences of gaps between adjacent Dyck paths appears to be unrelated to this sequence.
First differs from A279286 (which is monotone) at a(19). - Omar E. Pol, Feb 08 2017
a(n) = d if the point (d,d) belongs to the first vertical-line-segment of exactly length n found in the main diagonal of the pyramid described in A245092 (starting from the top). The diagram of the symmetries of sigma is also the top view of the pyramid. - Omar E. Pol, Feb 09 2017

			The four examples listed in A279286 are also examples for this sequences.
a(20) = 18915 is in the sequence since it is the first time that exactly 20 Dyck paths meet on the diagonal though a concurrence of exactly 19 paths on the diagonal happens only later at a(19) = 19818.

Cf. A128605, A237593, A240542, A245092, A279286.

a240542[n_] := Sum[(-1)^(k+1)*Ceiling[(n+1)/k - (k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
(* parameter cL must be sufficiently large for bound b *)
a282197[cL_, b_] := Module[{centers=Map[0&, Range[cL]], acc={1}, k=2, cPrev=1, cCur, len}, While[k<=b, cCur=a240542[k]; If[Last[acc]==cCur, AppendTo[acc,cCur], len=Length[acc]; If[centers[[len]]==0, centers[[len]]=cPrev]; acc={cCur}; cPrev=cCur]; k++]; centers]
a282197[50, 5000000] (* data *)
(* list processing implementation useful for "small" arguments only *)
a282197F[n_] := Map[Last, Sort[Normal[Map[First[First[#]]&, GroupBy[Split[Map[a240542, Range[n]]], Length[#]&]]]]]
a282197F[50000] (* computes a(1) .. a(20) *)

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.

A338535 a(n) is the smallest number k for which the width n at the diagonal equals the maximum width of the symmetric representation of sigma(k).

Original entry on oeis.org

1, 6, 72, 120, 3528, 840, 3600, 2520, 28800, 10080, 129600, 36960, 259200, 50400, 176400, 83160, 352800, 138600

Offset: 1

Hartmut F. W. Hoft, Nov 01 2020

All numbers computed so far for this sequence have a symmetric representation of sigma that consists of a single region.

Additional values computed through 2000000 are a(20,21,22,24,26,30) = (277200, 1411200, 360360, 960960, 942480, 1884960).

			a(3) = 72 = 2^3*3^2 is in the sequence since it is the smallest with maximum width 3 and width 3 at the diagonal for its symmetric representation of sigma. Number 60 is the smallest number with maximum width 3, but it has width 2 at the diagonal (see also Ax22222). a(3) has the following pattern for the widths of the legs up to the diagonal (see A249223): 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3 for its single region.
a(5) = 3528 = 2^3*3^2*7^2 is in the sequence, but does not have a monotone increase of widths up to the diagonal in its single region: 1...2...3...4...3...4...3...4...5 (there are 83 legs to the diagonal).

Cf. A128605, A235791, A237048, A237591, A237593, A249223, A250071.

(* Functions row[] and a237048[] are defined in A237048 *)
widthQ0[n_] := Module[{r=row[n], cW=0, mW=0, k}, For[k=1, k<=r, k++, cW+=(-1)^(k+1) a237048[n, k]; If[cW>mW, mW=cW]]; If[mW==cW, cW, 0]]
a338535[n_, b_] := Module[{list=Table[0, {b}], k, wQ}, For[k=1, k<=n, k++, wQ=widthQ0[k]; If[wQ!=0&&list[[wQ]]==0, list[[wQ]]=k]]; list]
Take[a338535[500000,25],18] (* sequence data *)

A319139 a(n) is the smallest position k >= 1 on the diagonal at which a record gap of size n between two adjacent Dyck paths of the symmetric representation of sigma starts.

Original entry on oeis.org

4, 50, 82, 1246, 581, 2494, 1744, 19961, 6981, 61136, 19210, 179669, 34935, 122268, 57628, 244539, 96062, 2415480, 192141, 978161, 249769, 1956341, 576404, 2200863, 499557

Offset: 1

Hartmut F. W. Hoft, Sep 11 2018

This sequence of positions of record gaps on the diagonal is not increasing, in contrast to the apparently increasing sequence A279286 of record numbers of Dyck paths jointly crossing the diagonal.
For n >= 2 it appears that a(2*n) > a(2*n+1), however a(2*n) < a(2*n+2) is false as a(12) = 179669 and a(14) = 122268 show, just as a(2n-1) < a(2*n+1) is false as a(23) = 576404 and a(25) = 499557 show.
Additional values of this sequence: a(27) = 1152829, a(29) = 999115, a(31) = 1498678, a(33) = 2305659.

			A240542(119) = 81 and A240542(120) = A240542(A128605(4)) = 85 establish the starting position on the diagonal of the first gap of size 3 as 82 = a(3).
A240542(3484799) = 2415479 and A240542(3484800) = A240542(A128605(19)) = 2415498 establish the starting position on the diagonal of the first gap of size 18 as 2415480 = a(18).

Cf. A128605, A237593, A240542, A279286.

a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[(Sqrt[8n+1]-1)/2]}]
(* parameter recs is the list of elements of the sequence in interval 1..m-1 already computed with an entry of 0 representing an element not yet found *)
a319139[m_, n_, recs_, ext_] := Module[{list=Join[recs, Table[0, ext]], a=a240542[m], i, b, g}, For[i=m+1, i<=n, i++, b=a240542[i]; g=b-a-1; If[g>0 && list[[g]]==0, list[[g]]=a+1]; a=b]; list]
a319139[1,3500000,{},40] (* data *)

It appears that a(n) = A240542(A128605(n+1)-1) + 1.

A354385 a(n) is the smallest odd number that has n middle divisors.

Original entry on oeis.org

1, 15, 1225, 2145, 99225, 17955, 893025, 51975, 4601025, 315315, 16769025, 855855, 12006225, 2567565, 108056025, 6531525, 385533225, 11486475, 225450225, 16787925, 1329696225, 38513475, 2701400625, 77702625, 6053618025, 80405325, 4846248225, 101846745, 2029052025, 218243025

Offset: 1

Hartmut F. W. Hoft, May 24 2022

nonn

This sequence is nonincreasing since a(5) > a(6), neither is the subsequence a(2n-1), n >= 1, of record odd counts of middle divisors since a(11) = 16769025 > 12006225 = a(13), nor is the subsequence a(2n), n >= 1, of record even counts since a(32) = 413377965 > 334639305 = a(34).
a(21) > 5*10^8.
Further computed values at even indices up to 5*10^8 are a(22, 24, 26, 28, 30, 32, 34) = (38513475, 77702625, 80405325, 101846745, 218243025, 413377965, 334639305).
Observation: At present all known terms >= a(4) are divisible by 3, all >= a(10) are divisible by 7, all >= a(12) are divisible by 11.
Conjecture: For every k, there is an n such that all >= a(n) are divisible by the first k odd primes.

			a(2) = 15 = A319529(3) is the smallest odd number with 2 middle divisors: 3 and 5.
a(3) = 1225 = A319529(116) is the smallest odd number with 3 middle divisors: 25, 35, and 45.

Omar E. Pol, Comments on A354385.

Cf. A067742, A016754, A128605, A235791, A237048, A237593, A245092, A249223, A319529, A320137.

middleDivC[n_] := Length[Select[Divisors[n], Sqrt[n/2]<=#=1&&list[[c]]==0, list[[c]]=k]]; list]
a354385[2*10^7, 20] (* long computation time *)

More terms from Amiram Eldar, Jun 07 2022

Edited by Omar E. Pol at the suggestion of N. J. A. Sloane, Jul 28 2022

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)).

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A282197 a(n) is the smallest number d if the point (d,d) is shared by exactly n different Dyck paths in the main diagonal of the diagram of the symmetries of sigma described in A237593.

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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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A338535 a(n) is the smallest number k for which the width n at the diagonal equals the maximum width of the symmetric representation of sigma(k).

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A319139 a(n) is the smallest position k >= 1 on the diagonal at which a record gap of size n between two adjacent Dyck paths of the symmetric representation of sigma starts.

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A354385 a(n) is the smallest odd number that has n middle divisors.

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