A319529 - OEIS

%I #44 Jul 31 2022 07:47:37

%S 1,9,15,25,35,45,49,63,77,81,91,99,117,121,135,143,153,165,169,187,

%T 195,209,221,225,231,247,255,273,285,289,299,315,323,325,345,357,361,

%U 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 Odd numbers that have middle divisors.

%C Odd numbers k such that the symmetric representation of sigma(k) has an odd number of parts.

%C From _Felix Fröhlich_, Sep 25 2018: (Start)

%C For the definition of middle divisors, see A067742.

%C Let t be a term of A005408. Then t is in this sequence iff A067742(t) != 0. (End)

%C From _Hartmut F. W. Hoft_, May 24 2022: (Start)

%C 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 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 Sequence A259417 is a subsequence of this sequence and of A320137 since an even power of a prime has a single middle divisor.

%C 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 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 .

%e .     _ _ _ _ _ 5

%e .    |_ _ _ _ _|

%e .              |_ _ 3

%e .              |_  |

%e .                |_|_ _ 5

%e .                    | |

%e .                    | |

%e .                    | |

%e .                    | |

%e .                    |_|

%e .

%t middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#<Sqrt[2n]&]

%t a319529[n_] := Select[Range[1, n, 2], middleDiv[#]!={}&]

%t a319529[651] (* _Hartmut F. W. Hoft_, May 24 2022 *)

%o (Python)

%o from itertools import islice, count

%o from sympy import divisors

%o def A319529_gen(startvalue=1): # generator of terms >= startvalue

%o     for k in count(max(1,startvalue+1-(startvalue&1)),2):

%o         if any((k <= 2*d**2 < 4*k for d in divisors(k,generator=True))):

%o             yield k

%o A319529_list = list(islice(A319529_gen(startvalue=11),40)) # _Chai Wah Wu_, Jun 09 2022

%Y Intersection of A005408 and A071562.

%Y For more information about the diagram see A237593.

%Y Cf. A000203, A067742, A071090, A236104, A237270, A237271, A239931, A239933, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319796, A319801, A319802.

%Y Cf. A016754, A237048, A249223, A259417, A320137.

%K nonn

%O 1,2

%A _Omar E. Pol_, Sep 23 2018