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A086886
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Ljungstrand's sequence: number of distinct solutions to n = (X + 1/x)(Y + 1/y), where x,y > 1 and X,Y are integers.
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1
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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
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1,3
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COMMENTS
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a(n) < n and lim_{n->inf} (Sum_{k=1..n} a(k)) / (n*log(n)^3) = 3/(2*Pi^2) (see article).
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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