

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.


1



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
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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[nk, X] != X  GCD[n/a + (nk)/X, k] != k, Continue[]]; Y = (nk)/X; x = (n/a + (nk)/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] (* JeanFrançois 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(nk, X)!=Xgcd(n/d+(nk)/X, k)!=k, next); Y=(nk)/X; x=(n/d+(nk)/X)/k; y=(d+X)/k; if(x==1y==1, next); if(X==Y, ant=ant+1/2, ant=ant+1)))); ant \\ Juliusz Brzezinski


CROSSREFS

Sequence in context: A291811 A291812 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



