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A319529
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Odd numbers that have middle divisors.
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11
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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
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OFFSET
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1,2
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COMMENTS
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Odd numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
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)
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)
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LINKS
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EXAMPLE
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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:
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MATHEMATICA
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middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#<Sqrt[2n]&]
a319529[n_] := Select[Range[1, n, 2], middleDiv[#]!={}&]
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PROG
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(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
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CROSSREFS
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For more information about the diagram see A237593.
Cf. A000203, A067742, A071090, A236104, A237270, A237271, A239931, A239933, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319796, A319801, A319802.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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