A354452 -id:A354452 - cp's OEIS frontend

A354451 Number of middle divisors of 2*n-1.

1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Omar E. Pol, May 30 2022

nonn

a(n) is the number of middle divisors of the n-th odd number.
a(n) is also the width of the terrace at the level 2*n-1 starting from the top in the main diagonal of the stepped pyramid described in A245092.
a(n) is also the number of central subparts in the symmetric representation of sigma(2n-1). For more information about the subparts see A279387.

Bisection of A067742.

Cf. A005408, A099774, A237048, A237270, A237271, A237593, A245092, A249351, A279387, A319529, A354452.

a[n_] := DivisorSum[2*n - 1, 1 &, n <= #^2 < 4*n - 2 &]; Array[a, 100] (* Amiram Eldar, Jun 01 2022 *)

a(n) = A067742(2n-1).

a(n) = A067742(A005408(n-1)).

A361208 Number of middle divisors of the n-th number whose divisors increase by a factor of 2 or less.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 1, 2, 2, 2, 4, 2, 1, 2, 2, 3, 2, 2, 2, 1, 2, 2, 4, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 1, 2, 3, 2, 2, 2, 4, 4, 2, 2, 3, 2, 2, 2, 2, 2, 2, 4

Offset: 1

Omar E. Pol, Mar 06 2023

nonn

The middle divisors of n are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).
Also consider the n-th number k with the property that the symmetric representation of sigma(k) has only one part. a(n) is the number of square cells on the axis of symmetry of the diagram.
For the diagrams related to the first 13 terms of this sequence see A317305.

Cf. A067742, A174973, A237048, A237270, A237593, A281007, A317305, A354452.

With[{$MaxExtraPrecision = 1000}, Map[Function[n, Count[Divisors[n], ?(Sqrt[n/2] <= # < Sqrt[2 n] &)]], Select[Range[500], And @@ Map[# <= 2 &, (Rest[#]/Most[#])] &@ Divisors[#] &]] ]  (* _Michael De Vlieger, Mar 28 2023, after Harvey P. Dale at A174973 *)

a(n) = A067742(A174973(n)).

cp's OEIS Frontend

A354451 Number of middle divisors of 2*n-1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula