A086886 Ljungstrand's sequence: number of distinct solutions to n = (X + 1/x)(Y + 1/y), where x,y > 1 and X,Y are integers.

0, 1, 2, 3, 4, 5, 5, 6, 8, 8, 8, 9, 8, 10, 14, 12, 10, 11, 10, 14, 17, 14, 13, 16, 18, 13, 17, 21, 13, 19, 12, 14, 23, 16, 26, 26, 14, 17, 21, 26, 16, 23, 16, 22, 30, 22, 18, 22, 24, 22, 26, 23, 18, 28, 33, 32, 29, 21, 20, 32, 19, 19, 30, 30, 35, 27, 18, 28, 31, 41, 20, 33, 19

Offset

1,3

Comments

a(n) < n and lim_{n->inf} (Sum_{k=1..n} a(k)) / (n*log(n)^3) = 3/(2*Pi^2) (see article).

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

J. Brzezinski, W. Holsztynski and P. Kurlberg, On the congruence ax+by=1 modulo xy, arXiv:math/0308194 [math.NT], 2003.

David A. Corneth, List of distinct solutions for a(n) where n = 1..300

Example

10 = (1+1/19)(9+1/2) = (3+1/13)(3+1/4) = (1+1/14)(9+1/3) = (3+1/8)(3+1/5) = (1+1/11)(9+1/6) = (1+1/5)(8+1/3) = (1+1/3)(7+1/2) = (1+1/10)(9+1/11). These are all 8 distinct such expressions for 10 so a(10) = 8. - David A. Corneth, Feb 18 2019

Mathematica

w[n_] := Module[{ant = 0}, Do[For[X = 1, X <= Floor[Sqrt[n]], X++, Do[If[ GCD[n-k, X] != X || GCD[n/a + (n-k)/X, k] != k, Continue[]]; Y = (n-k)/X; x = (n/a + (n-k)/X)/k; y = (a+X)/k; If[x == 1 || y == 1, Continue[]]; If[ X == Y, ant = ant+1/2, ant = ant+1], {k, Divisors[a+X]}]], {a, Divisors[n] }]; ant]; Array[w, 73] (* Jean-François Alcover, Feb 18 2019, translated from PARI *)

Prog

(PARI) a(n)=ant=0; fordiv(n, d, for(X=1, floor(sqrt(n)), fordiv(d+X, k, if(gcd(n-k, X)!=X||gcd(n/d+(n-k)/X, k)!=k, next); Y=(n-k)/X; x=(n/d+(n-k)/X)/k; y=(d+X)/k; if(x==1||y==1, next); if(X==Y, ant=ant+1/2, ant=ant+1)))); ant \\ Juliusz Brzezinski

Crossrefs

Sequence in context: A363694 A330292 A017866 * A017840 A017855 A051598

Adjacent sequences: A086883 A086884 A086885 * A086887 A086888 A086889

Keyword

nonn

Author

Ralf Stephan, Aug 22 2003; revised Dec 08 2004

Status

approved