A299777 -id:A299777 - cp's OEIS frontend

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.

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.

A361824 Sum of odd middle divisors of n, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

Original entry on oeis.org

1, 1, 0, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 8, 0, 0, 3, 0, 5, 0, 0, 0, 0, 5, 0, 0, 7, 0, 5, 0, 0, 0, 0, 12, 0, 0, 0, 0, 5, 0, 7, 0, 0, 14, 0, 0, 0, 7, 5, 0, 0, 0, 9, 0, 7, 0, 0, 0, 0, 0, 0, 16, 0, 0, 11, 0, 0, 0, 7, 0, 9, 0, 0, 0, 0, 18, 0, 0, 0, 9, 0, 0, 7, 0, 0, 0, 11, 0, 9, 20, 0, 0, 0, 0, 0, 0, 7, 20, 0

Offset: 1

Omar E. Pol, Mar 25 2023

Sum of odd divisors of n in the half-open interval [sqrt(n/2), sqrt(n*2)) (cf. A067742).

Also sum of odd numbers in the n-th row of A299761.

			For n = 8 the middle divisor of 8 is [2]. There are no odd middle divisors of 8 so a(8) = 0.
For n = 12 the middle divisors of 12 are [3, 4]. There is only one odd middle divisor of 12 so a(12) = 3.
For n = 15 the middle divisors of 15 are [3, 5]. There are two odd middle divisors of 15 so a(15) = 3 + 5 = 8.

Winston de Greef, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.
Michael De Vlieger, Detail of scalar scatterplot of a(n), n = 1..2^16, a(n) <= 384.

Cf. A000203, A000593, A067742, A071090, A071562, A299761, A299777, A303297, A358434, A361879.

Table[Total@ Select[Divisors[n], And[Sqrt[n/2] <= # < Sqrt[2 n], OddQ[#] ] &], {n, 100}] (* Michael De Vlieger, Mar 27 2023 *)

a(n) = vecsum(select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2)) && x%2), divisors(n))); \\ Michel Marcus, Mar 26 2023

a(n) = A071090(n) - A361879(n).

A361879 Sum of even middle divisors of n, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 10, 0, 0, 0, 4, 0, 6, 0, 4, 0, 0, 0, 6, 0, 0, 0, 8, 0, 6, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 6, 0, 8, 0, 0, 0, 16, 0, 0, 0, 8, 0, 6, 0, 0, 0, 10, 0, 14, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 12, 0, 0, 0, 8, 0, 10, 0, 0, 0, 0, 0, 20

Offset: 1

Omar E. Pol, Mar 27 2023

Sum of even divisors of n in the half-open interval [sqrt(n/2), sqrt(n*2)).

Also sum of even numbers in the n-th row of A299761.

			For n = 18 the middle divisor of 18 is [3]. There are no even middle divisors of 18 so a(18) = 0.
For n = 20 the middle divisors of 20 are [4, 5]. There is only one even middle divisor of 20 so a(20) = 4.
For n = 24 the middle divisors of 24 are [4, 6]. There are two even middle divisors of 24 so a(24) = 4 + 6 = 10.

Robert Israel, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^16 (ignoring zeros).

Cf. A000203, A067742, A071090, A071562, A146076, A299761, A299777, A303297, A358434, A361561, A361824.

f:= proc(n) local D;
     if n::odd then return 0 fi;
     D:= select(proc(d) local s; if d::odd then return false fi; s:= d^2; s >= n/2 and s < 2*n end proc, numtheory:-divisors(n)); convert(D,`+`) end proc:
map(f, [$1..100]); # Robert Israel, Mar 18 2024

Table[DivisorSum[n, # &, And[EvenQ[#], Sqrt[n/2] <= # < Sqrt[2 n]] &], {n, 120}] (* Michael De Vlieger, Mar 28 2023 *)

a(n) = vecsum(select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2)) && !(x%2)), divisors(n))); \\ Michel Marcus, Mar 31 2023

a(n) = A071090(n) - A361824(n).

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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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A361824 Sum of odd middle divisors of n, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

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A361879 Sum of even middle divisors of n, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

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