A219695 For odd numbers 2n - 1, half the difference between the largest divisor not exceeding the square root, and the least divisor not less than the square root.

0, 1, 2, 3, 0, 5, 6, 1, 8, 9, 2, 11, 0, 3, 14, 15, 4, 1, 18, 5, 20, 21, 2, 23, 0, 7, 26, 3, 8, 29, 30, 1, 4, 33, 10, 35, 36, 5, 2, 39, 0, 41, 6, 13, 44, 3, 14, 7, 48, 1, 50, 51, 4, 53, 54, 17, 56, 9, 2, 5, 0, 19, 10, 63, 20, 65, 6, 3, 68, 69, 22, 1, 12, 7, 74, 75, 4, 13, 78, 25, 8, 81, 2, 83, 0, 5, 86, 9, 28, 89

Offset

1,3

Comments

We consider 2n-1 which has only odd divisors, so any difference among them is always even.

a(n) = 0 if and only if 2n - 1 is a square.

From Charles Kusniec, Oct 31 2024: (Start)

a(n) = 1 iff 2n is a square.

a(n) = 2 iff 2n-1 = (2k-1)^2 - 4.

a(n) = 3 iff 2n-1 = (2k)^2 - 9.

Since 2n-1 is odd, 2n-1 = d*D can be written 2n-1 = (x-y)*(x+y) = x^2 - y^2 with integers x and y, from which a(n) is the smallest possible y, which also means smallest possible x and that being x = A377499(n).

Record values occur when 2n-1 is an odd prime (n in A006254), and then a(n) = n-1 (and A377499(n) = n). (End)

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) = (A056737(2n-1))/2 = (A033677(2n-1) - A033676(2n-1))/2.

a(n) = sqrt(A377499(n)^2 - (2n-1)). - Charles Kusniec, Oct 31 2024

Example

For n = 2, consider divisors of 2n - 1 = 3 which are {1, 3}. The least one greater than or equal to sqrt(3) is 3, the largest one less than or equal to sqrt(3) is 1; whence a(2) = (3 - 1)/2 = 1.
For n = 14, consider divisors of 2n - 1 = 27 which are {1, 3, 9, 27}. The least one greater than or equal to sqrt(27) is 9, the largest one less than or equal to sqrt(27) is 3; whence a(14) = (9 - 3)/2 = 3.
For n = 1, 5, 13, 25, ..., the number 2n - 1 equals the square 1, 9, 25, 49, ...; so the two beforementioned "median divisors" coincide with the square root, and a(n) = 0/2 = 0.

Mathematica

Table[(Divisors[n][[(Length[Divisors[n]] - Boole[IntegerQ[Sqrt[n]]])/2 + 1]] - Divisors[n][[(Length[Divisors[n]] + Boole[IntegerQ[Sqrt[n]]])/2]])/2, {n, 1, 199, 2}] (* Alonso del Arte, Nov 25 2012, corrected March 21 2024, with help from Giorgos Kalogeropoulos *)
A219695[n_] := (d = Divisors[2n - 1]; l = Floor[Length@d/2 + 1]; (d[[l]] - d[[-l]])/2); Array[A219695, 100] (* Giorgos Kalogeropoulos, Mar 15 2024 *)

Prog

(PARI)
A056737(n)={n=divisors(n); n[(2+#n)\2]-n[(1+#n)\2]}
A219695(n)=A056737(2*n-1)/2  \\ M. F. Hasler, Nov 25 2012

Crossrefs

Cf. A033676, A033677, A056737, A001844, A016754, A377499.

Cf. A006254 (indices of record highs).

Sequence in context: A351078 A128214 A307865 * A364575 A267186 A248092

Adjacent sequences: A219692 A219693 A219694 * A219696 A219697 A219698

Keyword

nonn

Author

M. F. Hasler, Nov 25 2012

Status

approved