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A346868
Sum of divisors of the numbers with no middle divisors.
8
4, 6, 8, 18, 12, 14, 24, 18, 20, 32, 36, 24, 42, 40, 30, 32, 48, 54, 38, 60, 56, 42, 44, 84, 72, 48, 72, 98, 54, 72, 80, 90, 60, 62, 96, 84, 68, 126, 96, 72, 74, 114, 124, 140, 168, 80, 126, 84, 108, 132, 120, 90, 168, 128, 144, 120, 98, 102, 216, 104, 192, 162, 108, 110
OFFSET
1,1
COMMENTS
The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is equal to zero.
So knowing this characteristic shape we can know if a number has middle divisors (or not) just by looking at the diagram, even ignoring the concept of middle divisors.
Therefore we can see a geometric pattern of the distribution of the numbers with no middle divisors in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
All terms are even numbers.
FORMULA
a(n) = A000203(A071561(n)).
EXAMPLE
a(4) = 18 because the sum of divisors of the fourth number with no middle divisors (i.e., 10) is 1 + 2 + 5 + 10 = 18.
On the other hand we can see that in the main diagonal of every diagram the width is equal to zero as shown below.
Illustration of initial terms:
m(n) = A071561(n).
.
n m(n) a(n) Diagram
. _ _ _ _ _ _ _ _ _ _ _ _ _
| | | | | | | | | | | | | | | | | | | | | |
_ _|_| | | | | | | | | | | | | | | | | | | | |
1 3 4 |_ _| _|_| | | | | | | | | | | | | | | | | | |
_ _ _| _|_| | | | | | | | | | | | | | | | |
2 5 6 |_ _ _| _| _ _| | | | | | | | | | | | | | | |
_ _ _ _| | _ _|_| | | | | | | | | | | | | |
3 7 8 |_ _ _ _| _ _|_| _ _|_| | | | | | | | | | | |
| _| | _ _ _| | | | | | | | | | |
_ _ _ _ _| | _|_| _ _ _|_| | | | | | | | |
4 10 18 |_ _ _ _ _ _| _ _| | _ _ _|_| | | | | | |
5 11 12 |_ _ _ _ _ _| | _| _| | _ _ _ _|_| | | | |
_ _ _ _ _ _ _| | _| _ _| | | _ _ _ _|_| | |
6 13 14 |_ _ _ _ _ _ _| | _ _| _| _| | | _ _ _ _ _| |
7 14 24 |_ _ _ _ _ _ _ _| | | | _|_| | _ _ _ _ _|
| _ _| _ _|_| | |
_ _ _ _ _ _ _ _ _| | _ _| _| _|_|
8 17 18 |_ _ _ _ _ _ _ _ _| | |_ _ _| |
_ _ _ _ _ _ _ _ _ _| | _ _| _|
9 19 20 |_ _ _ _ _ _ _ _ _ _| | | _ _|
_ _ _ _ _ _ _ _ _ _ _| | _ _ _|
10 21 32 |_ _ _ _ _ _ _ _ _ _ _| | | _ _|
11 22 36 |_ _ _ _ _ _ _ _ _ _ _ _| | |
12 23 24 |_ _ _ _ _ _ _ _ _ _ _ _| | |
| |
_ _ _ _ _ _ _ _ _ _ _ _ _| |
13 26 42 |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
MATHEMATICA
s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], 0, Plus @@ d]]; Select[Array[s, 110], # > 0 &] (* Amiram Eldar, Aug 19 2021 *)
PROG
(PARI) is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1; \\ A071561 apply(sigma, select(is, [1..150])) \\ Michel Marcus, Aug 19 2021
CROSSREFS
Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors).
Sequence in context: A112160 A132040 A210459 * A279896 A247280 A226237
KEYWORD
nonn
AUTHOR
Omar E. Pol, Aug 18 2021
STATUS
approved