cp's OEIS Frontend

A319529 Odd numbers that have middle divisors.

%I A319529 #44 Jul 31 2022 07:47:37
%S A319529 1,9,15,25,35,45,49,63,77,81,91,99,117,121,135,143,153,165,169,187,
%T A319529 195,209,221,225,231,247,255,273,285,289,299,315,323,325,345,357,361,
%U A319529 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
%N A319529 Odd numbers that have middle divisors.
%C A319529 Odd numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
%C A319529 From _Felix Fröhlich_, Sep 25 2018: (Start)
%C A319529 For the definition of middle divisors, see A067742.
%C A319529 Let t be a term of A005408. Then t is in this sequence iff A067742(t) != 0. (End)
%C A319529 From _Hartmut F. W. Hoft_, May 24 2022: (Start)
%C A319529 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)).
%C A319529 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.
%C A319529 Sequence A259417 is a subsequence of this sequence and of A320137 since an even power of a prime has a single middle divisor.
%C A319529 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)
%e A319529 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:
%e A319529 .
%e A319529 .     _ _ _ _ _ 5
%e A319529 .    |_ _ _ _ _|
%e A319529 .              |_ _ 3
%e A319529 .              |_  |
%e A319529 .                |_|_ _ 5
%e A319529 .                    | |
%e A319529 .                    | |
%e A319529 .                    | |
%e A319529 .                    | |
%e A319529 .                    |_|
%e A319529 .
%t A319529 middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#<Sqrt[2n]&]
%t A319529 a319529[n_] := Select[Range[1, n, 2], middleDiv[#]!={}&]
%t A319529 a319529[651] (* _Hartmut F. W. Hoft_, May 24 2022 *)
%o A319529 (Python)
%o A319529 from itertools import islice, count
%o A319529 from sympy import divisors
%o A319529 def A319529_gen(startvalue=1): # generator of terms >= startvalue
%o A319529     for k in count(max(1,startvalue+1-(startvalue&1)),2):
%o A319529         if any((k <= 2*d**2 < 4*k for d in divisors(k,generator=True))):
%o A319529             yield k
%o A319529 A319529_list = list(islice(A319529_gen(startvalue=11),40)) # _Chai Wah Wu_, Jun 09 2022
%Y A319529 Intersection of A005408 and A071562.
%Y A319529 For more information about the diagram see A237593.
%Y A319529 Cf. A000203, A067742, A071090, A236104, A237270, A237271, A239931, A239933, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319796, A319801, A319802.
%Y A319529 Cf. A016754, A237048, A249223, A259417, A320137.
%K A319529 nonn
%O A319529 1,2
%A A319529 _Omar E. Pol_, Sep 23 2018