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A319529
Odd numbers that have middle divisors.
11
1, 9, 15, 25, 35, 45, 49, 63, 77, 81, 91, 99, 117, 121, 135, 143, 153, 165, 169, 187, 195, 209, 221, 225, 231, 247, 255, 273, 285, 289, 299, 315, 323, 325, 345, 357, 361, 375, 391, 399, 405, 425, 435, 437, 441, 459, 475, 483, 493, 513, 525, 527, 529, 551, 561, 567, 575, 589, 609, 621, 625, 627, 651
OFFSET
1,2
COMMENTS
Odd numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
From Felix Fröhlich, Sep 25 2018: (Start)
For the definition of middle divisors, see A067742.
Let t be a term of A005408. Then t is in this sequence iff A067742(t) != 0. (End)
From Hartmut F. W. Hoft, May 24 2022: (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)) on the diagonal which equals the triangle entry A249223(n, A003056(n)).
All terms in sequence A016754 have an odd number of middle divisors, forming a subsequence of this sequence; A016754(18) = a(116) = 1225 = 5^2 * 7^2 is the smallest number in A016754 with 3 middle divisors: 25, 35, 49.
Sequence A259417 is a subsequence of this sequence and of A320137 since an even power of a prime has a single middle divisor.
The maximum widths of the center part of the symmetric representation of sigma(a(n)), SRS(a(n)), need not occur at the diagonal. For example, a(304) = 3^3 * 5^3 = 3375, SRS(3375) has 3 parts, its center part has maximum width 3 while its width at the diagonal equals 2 = A067742(3375), and divisors 45 and 75 are the two middle divisors of a(304). (End)
EXAMPLE
9 is in the sequence because it's an odd number and the symmetric representation of sigma(9) = 13 has an odd number of parts (more exactly three parts), as shown below:
.
. _ _ _ _ _ 5
. |_ _ _ _ _|
. |_ _ 3
. |_ |
. |_|_ _ 5
. | |
. | |
. | |
. | |
. |_|
.
MATHEMATICA
middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#<Sqrt[2n]&]
a319529[n_] := Select[Range[1, n, 2], middleDiv[#]!={}&]
a319529[651] (* Hartmut F. W. Hoft, May 24 2022 *)
PROG
(Python)
from itertools import islice, count
from sympy import divisors
def A319529_gen(startvalue=1): # generator of terms >= startvalue
for k in count(max(1, startvalue+1-(startvalue&1)), 2):
if any((k <= 2*d**2 < 4*k for d in divisors(k, generator=True))):
yield k
A319529_list = list(islice(A319529_gen(startvalue=11), 40)) # Chai Wah Wu, Jun 09 2022
CROSSREFS
Intersection of A005408 and A071562.
For more information about the diagram see A237593.
Sequence in context: A176404 A227198 A279102 * A249730 A342418 A251415
KEYWORD
nonn
AUTHOR
Omar E. Pol, Sep 23 2018
STATUS
approved