A071090 -id:A071090 - cp's OEIS frontend

A071562 Numbers n such that the sum of the middle divisors of n (A071090) is not zero.

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 66, 70, 72, 77, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 104, 108, 110, 112, 117, 120, 121, 126, 128, 130, 132, 135, 140, 143, 144, 150, 153, 154, 156, 160

Offset: 1

Robert G. Wilson v, May 30 2002

nonn

Numbers n such that A067742(n) is nonzero.
Numbers of the form m*k with m <= k <= 2m. - Vladeta Jovovic, May 07 2005
Numbers occurring in A100345 (except 0). - Franklin T. Adams-Watters, Apr 04 2010
This sequence is closed under multiplication. If n = a*b with a <= b <= 2a, and m = c*d with c <= d <= 2c, then min(a*d,b*c)*max(a*d,b*c) is a factorization of m*n with the specified property. - Franklin T. Adams-Watters, Apr 07 2010
Also numbers n with the property that the number of parts in the symmetric representation of sigma(n) is odd. - Michel Marcus and Omar E. Pol, Apr 25 2014. (For a proof see the link in A071561.) - Hartmut F. W. Hoft, Sep 09 2015
Among these numbers, those with sigma(n) also odd are 1, 2, 4, 8, 9, 16, ..., that is, probably A028982 (squares and twice squares). - Michel Marcus, Jun 21 2014
Records in A244367. - Omar E. Pol, Jul 27 2014
Starting from a(5), the sequence is a subset of a sequence formed out of the prime factorization of A129912(m), m >= 5; see associated PARI code in Prog section. - Bill McEachen, Jan 25 2018
For numbers n = 2^m * q, m >= 0, q odd, and where r(n) = floor( (sqrt(8n+1) - 1)/2 ), the symmetric representation of sigma(n) has an odd number of parts precisely when there exists an odd divisor d of n satisfying d <= r(n) and d*2^(m+1) > r(n); see the link for a proof and see the associated Mathematica code. - Hartmut F. W. Hoft, Feb 12 2018
All hexagonal numbers A000384 > 0 are in the sequence. - Omar E. Pol, Aug 28 2018

			From _Hartmut F. W. Hoft_, Feb 12 2018: (Start)
63 = 3^2*7 is in the sequence since 7*2^1 > r(63) = 10.
80 = 2^4*5 is in the sequence since 1*2^5 > r(80) = 12. (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Hartmut F. W. Hoft, Proof of property for odd divisors
H. Maier and G. Tenenbaum, On the set of divisors of an integer, Invent. Math. 76:1 (1984), 121-128.

Cf. A067742.
The complement is A071561.
Cf. A000203, A028982, A071090, A100345, A175040, A176039 (primitive elements), A237270, A237271, A237593, A244367.

f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[175], f[ # ] != 0 &]
(* Related to the symmetric representation of sigma *)
(* subsequence of odd parts of number k for m <= k <= n *)
(* Function a237270[] is defined in A237270 *)
(* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *)
a071562[m_,n_]:=Select[Range[m,n],OddQ[Length[a237270[#]]]&]
a071562[1,160] (* data *)
(* Hartmut F. W. Hoft, Jun 23 2014 *)
(* implementation using the odd divisor property *)
evenExp[n_] := First[NestWhile[{#[[1]]+1, #[[2]]/2}&, {0, n}, EvenQ[Last[#]]&]]
oddSRQ[n_] := Module[{e=2^evenExp[n], Floor[(Sqrt[8n+1]-1)/2]}, Select[Divisors[n/e], #<=r&&2 e #>r&]!={}]
a071562D[m_, n_] := Select[Range[m, n], oddSRQ]
a071562D[1, 160] (* data *) (* Hartmut F. W. Hoft, Feb 12 2018 *)

is(n)=fordiv(n,d, if(d^2>=n/2 && d^2<2*n, return(1))); 0 \\ Charles R Greathouse IV, Aug 01 2016

is(n,f=factor(n))=my(t=(n+1)\2); fordiv(f,d, if(d^2>=t, return(d^2<2*n))); 0 \\ Charles R Greathouse IV, Jan 22 2018

list(lim)=my(v=List(),t); forfactored(n=1,lim\1, t=(n[1]+1)\2; fordiv(n[2],d, if(d^2>=t, if(d^2<2*n[1], listput(v,n[1])); break))); Vec(v) \\ Charles R Greathouse IV, Jan 22 2018

/* functional code associated to the A129912 comment above */ for(j5=5, length(A129912), a=Mat(); a=factor(A129912[j5]); sum2=0; for(i5=1, length(a[,2]), sum2=sum2+a[i5,2]); listput(final,length(a[,1])*sum2)); v=Set(final); \\ Bill McEachen, Jan 25 2018

A071561 Numbers with no middle divisors (cf. A071090).

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 111, 113, 114

Offset: 1

Robert G. Wilson v, May 30 2002

nonn

Numbers k such that A071090(k) is 0.
Conjecture: lim_{n->oo} a(n)/n = 4/3.
Regarding the above conjecture, numerical calculations suggest that this limit is smaller than 4/3. See A071540. - Amiram Eldar, Jul 27 2024
Also numbers n with the property that the number of parts in the symmetric representation of sigma(n) is even. - Michel Marcus and Omar E. Pol, Apr 25 2014 [For a proof see the link. - Hartmut F. W. Hoft, Sep 09 2015]
Middle divisors are divisors d with sqrt(k/2) <= d < sqrt(2k). - Michael B. Porter, Oct 19 2018

			From _Michael B. Porter_, Oct 19 2018: (Start)
The divisors of 21 are 1, 3, 7, and 21.  Since none of these are between sqrt(21/2) = 3.24... and sqrt(2*21) = 6.48..., 21 is in the sequence.
The divisors of 20 are 1, 2, 4, 5, 10, and 20.  Since 4 and 5 are both between sqrt(20/2) = 3.16... and sqrt(2*20) = 6.32..., 20 is not in the sequence. (End)

Iain Fox, Table of n, a(n) for n = 1..10000
Hartmut F. W. Hoft, On the symmetric spectrum of odd divisors of a number, (2015), (Note that in this paper, A241561 should be replaced with A071561, and A241562 should be replaced with A071562. Also note that "the symmetric spectrum of odd divisors of a number" seems to be an attempt to call with a new name to a diagram known since 2014 as "the symmetric representation of sigma(n)"). - _Omar E. Pol_, Oct 08 2018
José Manuel Rodríguez Caballero, Elementary number-theoretical statements proved by Language Theory, arXiv:1709.09617 [math.LO], 2017.
J. M. Rodríguez Caballero, Symmetric Dyck Paths and Hooley's Δ-Function, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.

Cf. A071090, A071540, A071562, A071563, A237048, A237270, A237271, A237593, A241008.

f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[125], f[ # ] == 0 &]
(* Related to the symmetric representation of sigma *)
(* subsequence of even parts of number k for m <= k <= n *)
(* Function a237270[] is defined in A237270 *)
(* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *)
a071561[m_, n_]:=Select[Range[m, n], EvenQ[Length[a237270[#]]]&]
a071561[1, 114] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)
Select[Range@ 120, Function[n, Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] == {}]] (* Michael De Vlieger, Jan 03 2017 *)

is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1 \\ Iain Fox, Dec 19 2017

is(n,f=factor(n))=my(t=(n+1)\2); fordiv(f,d, if(d^2>=t, return(d^2>2*n))); 0 \\ Charles R Greathouse IV, Jan 22 2018

list(lim)=my(v=List(),t); forfactored(n=3,lim\1, t=(n[1]+1)\2; fordiv(n[2],d, if(d^2>=t, if(d^2>2*n[1], listput(v,n[1])); break))); Vec(v) \\ Charles R Greathouse IV, Jan 22 2018

A071563 Least k such that there are no middle divisors of k (A071090) through k+n.

Original entry on oeis.org

3, 10, 21, 73, 145, 426, 533, 652, 1335, 2669, 2929, 6513, 6917, 8401, 8859, 10012, 25351, 27281, 27281, 87054, 113051, 122851, 122851, 122851, 122851, 178569, 178569, 185506, 185506, 599206, 599206, 599206, 1308163, 1888133, 1888133, 1888133, 1888133, 1888133

Offset: 0

Robert G. Wilson v, May 30 2002

nonn

			a(3) = 73 because 73, 74, 75 & 76 are present in A071562.

Cf. A071090, A071561, A071562.

a(18) inserted and more terms from Sean A. Irvine, Jul 26 2024

A071547 Determinant of the n X n matrix whose element (i,j) equals f(|i-j|) where f(n) is the sum of middle divisors (A071090).

Original entry on oeis.org

0, -1, 2, 0, -2, 3, 8, -259, -9006, 37575, 1988916, 41615129, -444915500, 4752358604, -50506657344, 532544416084, -4450492081868, -10234691885873, 241215169689690, 21696729746925312, 123619114295636184, 174381272189413488, -43909101832286346168, -439142701170244441024

Offset: 1

Robert G. Wilson v, May 30 2002

sign

Cf. A071090.

f[n_] := If[n > 0, Plus @@ Select[Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &], 0]; Table[ Det[ Table[ f[ Abs[i - j]], {i, 1, n}, {j, 1, n}]], {n, 1, 25}]

A071548 Determinant of the n X n matrix whose element (i,j) equals f(|i-j|) where f(n) is 1 if the sum of middle divisors (A071090) > 0, else 0.

Original entry on oeis.org

0, -1, 2, 0, 0, 0, 2, 1, -4, 12, 0, -12, -4, 7, 44, 48, 0, -48, 44, 345, 334, -1196, 2328, 4796, 8022, -35417, 123840, -118503, 89082, 48160, -414248, 2107104, -2358342, -27802725, -168156872, 415198413, 804518348, -2450715149, -711441100, 3648319725, -2721771200, -17289389524, 205130688

Offset: 1

Robert G. Wilson v, May 30 2002

sign

Cf. A071090.

f[n_] := If[n > 0, If[Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &] == 0, 0, 1], 0]; Table[ Det[ Table[ f[ Abs[i - j]], {i, 1, n}, {j, 1, n}]], {n, 1, 50}]

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

Original entry on oeis.org

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

A299761 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which row n lists the middle divisors of n, or 0 if there are no middle divisors of n.

Original entry on oeis.org

1, 1, 0, 2, 0, 2, 3, 0, 2, 3, 0, 0, 3, 4, 0, 0, 3, 5, 4, 0, 3, 0, 4, 5, 0, 0, 0, 4, 6, 5, 0, 0, 4, 7, 0, 5, 6, 0, 4, 0, 0, 5, 7, 6, 0, 0, 0, 5, 8, 0, 6, 7, 0, 0, 5, 9, 0, 0, 6, 8, 7, 5, 0, 0, 0, 6, 9, 0, 7, 8, 0, 0, 0, 6, 10, 0, 0, 7, 9, 8, 0, 6, 11, 0, 0, 0, 7, 10, 0, 6, 8, 9, 0, 0, 0, 0, 7, 11, 0, 0, 8, 10

Offset: 1

Omar E. Pol, Jun 08 2018

The middle divisors of n are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).

			Triangle begins (rows 1..16):
1;
1;
0;
2;
0;
2, 3;
0;
2;
3;
0;
0;
3, 4;
0;
0;
3, 5;
4;
...
For n = 6 the middle divisors of 6 are 2 and 3, so row 6 is [2, 3].
For n = 7 there are no middle divisors of 7, so row 7 is [0].
For n = 8 the middle divisor of 8 is 2, so row 8 is [2].
For n = 72 the middle divisors of 72 are 6, 8 and 9, so row 72 is [6, 8, 9].

Michael De Vlieger, Table of n, a(n) for n = 1..14002 (rows 1 <= n <= 10^4)

Row sums give A071090.
The number of nonzero terms in row n is A067742(n).
Nonzero terms give A303297.
Indices of the rows where there are zeros give A071561.
Indices of the rows where there are nonzero terms give A071562.
Cf. A027750, A281007, A299777.

Table[Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] /. {} -> {0}, {n, 80}] // Flatten (* Michael De Vlieger, Jun 14 2018 *)

row(n) = my(v=select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2))), divisors(n))); if (#v, v, [0]); \\ Michel Marcus, Aug 04 2022

A303297 List of middle divisors: for every positive integer that has middle divisors, add its middle divisors to the sequence.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 5, 4, 3, 4, 5, 4, 6, 5, 4, 7, 5, 6, 4, 5, 7, 6, 5, 8, 6, 7, 5, 9, 6, 8, 7, 5, 6, 9, 7, 8, 6, 10, 7, 9, 8, 6, 11, 7, 10, 6, 8, 9, 7, 11, 8, 10, 9, 7, 12, 8, 11, 9, 10, 7, 13, 8, 12, 7, 9, 11, 10, 8, 13, 9, 12, 10, 11, 8, 14, 9, 13, 8, 10, 12, 15, 11, 9, 14, 8, 10, 13, 11, 12, 9, 15

Offset: 1

Omar E. Pol, Apr 30 2018

The middle divisors of k (see A299761) are the divisors in the half-open interval [sqrt(k/2), sqrt(k*2)), k >= 1.

			The middle divisor of 1 is 1, so a(1) = 1.
The middle divisor of 2 is 1, so a(2) = 1.
There are no middle divisors of 3.
The middle divisor of 4 is 2, so a(3) = 2.
There are no middle divisors of 5.
The middle divisors of 6 are 2 and 3, so a(4) = 2 and a(5) = 3.
There are no middle divisors of 7.
The middle divisor of 8 is 2, so a(6) = 2.
The middle divisor of 9 is 3, so a(7) = 3.
There are no middle divisors of 10.
There are no middle divisors of 11.
The middle divisors of 12 are 3 and 4, so a(8) = 3 and a(9) = 4.

Michel Marcus, Table of n, a(n) for n = 1..6934
Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 2^16.
Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 345, labeling n, and showing composite prime powers in gold, squarefree composites in green, numbers neither squarefree nor composite in blue, and highlighting products of composite prime powers in large light blue.
Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 2^16, with same color function as above so as to show patterns according to prime power decomposition of n.

Concatenate the nonzero rows of A299761 (that is, the nonzero terms of A299761).

Cf. A027750, A067742, A071090, A071562, A281007, A299777.

Table[Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] /. {} -> Nothing, {n, 135}] // Flatten (* Michael De Vlieger, Jun 14 2018 *)

lista(nn) = {my(list = List()); for (n=1, nn, my(v = select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2))), divisors(n))); for (i=1, #v, listput(list, v[i]));); Vec(list);} \\ Michel Marcus, Mar 26 2023

A319796 Even numbers that have middle divisors, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 50, 54, 56, 60, 64, 66, 70, 72, 80, 84, 88, 90, 96, 98, 100, 104, 108, 110, 112, 120, 126, 128, 130, 132, 140, 144, 150, 154, 156, 160, 162, 168, 170, 176, 180, 182, 190, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 238, 240, 242, 252, 256

Offset: 1

Omar E. Pol, Sep 28 2018

nonn

Even numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
An even number A005843 is in this sequence iff A067742(t) != 0.
For the definition of middle divisors, see A067742.
For more information about the symmetric representation of sigma(k) see A237593.
From Hartmut F. W. Hoft, Mar 28 2023: (Start)
By Theorem 1 (iii) in A067742, the number of middle divisors of a(n) equals the width of the symmetric representation of sigma(a(n)), SRS(a(n)), on the diagonal which equals the triangle entry A249223(n, A003056(n)). The maximum widths of the center part of SRS(a(n)) need not occur at the diagonal.
For example, a(7) = 2 * 3^2 = 18, SRS(18) has a single part with maximum width 2 while its width at the diagonal equals 1 = A067742(18), and divisor 3 is the only middle divisor of a(7). (End)

			6 is in the sequence because it's an even number and the symmetric representation of sigma(6) = 12 has an odd number of parts (more exactly only one part), as shown below:
.    _ _ _ _
.   |_ _ _  |_ 12
.         |   |_
.         |_ _  |
.             | |
.             | |
.             |_|
.
Also 50 is in the sequence because it's an even number and the symmetric representation of sigma(50) = 93 has an odd number of parts (more exactly three parts), they are [39, 15, 39].
a(34) = 110 = 2 * 5 * 11 has 10 and 11 as its middle divisors, and SRS(a(34)) has 3 parts and width 2 at the diagonal. -  _Hartmut F. W. Hoft_, Mar 28 2023

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Intersection of A005843 and A071562.

Cf. A000203, A067742, A071090, A236104, A237270, A237271, A237593, A239932, A239934, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319801, A319802.

filter:= n -> ormap(t -> t^2 >= n/2 and t^2 < 2*n, numtheory:-divisors(n)):
select(filter, 2*[$1..1000]); # Robert Israel, Mar 29 2023

middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#Hartmut F. W. Hoft, Mar 28 2023 *)

Name clarified by Omar E. Pol, Mar 28 2023

A299777 a(n) is the sum of middle divisors of the n-th number that has middle divisors.

Original entry on oeis.org

1, 1, 2, 5, 2, 3, 7, 8, 4, 3, 9, 10, 5, 11, 11, 4, 12, 6, 13, 13, 14, 14, 7, 5, 15, 15, 16, 16, 8, 17, 17, 23, 18, 18, 9, 19, 19, 19, 20, 20, 7, 20, 10, 21, 21, 21, 22, 22, 45, 11, 23, 23, 23, 24, 24, 24, 37, 25, 26, 25, 25, 26, 26, 26, 13, 27, 27, 55, 27, 28, 29, 28, 28, 14, 29, 10, 29, 29, 30, 29

Offset: 1

Omar E. Pol, Apr 07 2018

nonn

The middle divisors of n, are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).

			For n = 12 the 12th number that has middle divisors is 24. The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The middle divisors of 24 are 4 and 6, and the sum of them is 4 + 6 = 10, so a(12) = 10.

Nonzero terms in A071090.
Companion of A281007.
Cf. A071562.

lista(nn) = for (n=1, nn, if (s=sumdiv(n, d, if(d^2>=n/2 && d^2<2*n, d, 0)), print1(s, ", "))); \\ Michel Marcus, Apr 24 2018

a(n) = A071090(A071562(n)).

cp's OEIS Frontend

A071562 Numbers n such that the sum of the middle divisors of n (A071090) is not zero.

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A071561 Numbers with no middle divisors (cf. A071090).

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A071563 Least k such that there are no middle divisors of k (A071090) through k+n.

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A071547 Determinant of the n X n matrix whose element (i,j) equals f(|i-j|) where f(n) is the sum of middle divisors (A071090).

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A071548 Determinant of the n X n matrix whose element (i,j) equals f(|i-j|) where f(n) is 1 if the sum of middle divisors (A071090) > 0, else 0.

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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)).

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A299761 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which row n lists the middle divisors of n, or 0 if there are no middle divisors of n.

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A303297 List of middle divisors: for every positive integer that has middle divisors, add its middle divisors to the sequence.

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