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Search: id:A106200
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| A106200 |
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a(n)=denominator of the probability that (x-y)/(x+y)+(y-z)/(y+z)+(z-u)/(z+u)+ (u-x)/(u+x) >0, assuming that each random quadruple of integers (x,y,z,u), with a<=x,y,z,u<=n, is equally likely. |
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+0
2
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| 1, 1, 27, 64, 125, 324, 2401, 512, 6561, 2500, 14641, 324, 28561, 2401, 50625, 8192, 83521, 8748, 130321, 1250, 194481, 14641, 279841, 82944, 390625, 114244, 531441, 153664, 707281, 202500, 923521, 262144, 1185921, 334084, 1500625, 209952
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Numerators form A106199.
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REFERENCES
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E. Deutsch and M. S. Klamkin, Problem 10540, Amer. Math. Monthly 108, (2001), p. 172.
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FORMULA
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a(n)=denominator of [n(n-1)(n^2-n-1)+4sum(floor(n/(k^2))*phi(k), k=2...floor(sqrt(n)))-2sum(floor(n/k)^2*phi(k), k=2... n)]/(2n^4).
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EXAMPLE
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a(3)=27 because at the 81 quadruples (x,y,z,u) (1<=x,y,z,u<=3) the function
(x-y)/(x+y)+(y-z)/(y+z)+(z-u)/(z+u)+(u-x)/(u+x) assumes twelve times the value 1/30, twelve times the value -1/30 and fiftyseven times the value 0; then the considered probablity is 12/81=4/27.
0,0,4/27,15/64,36/125,103/324,832/2401
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MAPLE
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with(numtheory): a:=n*(n-1)*(n^2-n-1): b:=4*sum(floor(n/k^2)*phi(k), k=2..floor(sqrt(n))): c:=2*sum((floor(n/k))^2*phi(k), k=2..n): p:=proc(n) (a+b-c)/2/n^4 end: seq(denom(simplify(p(n))), n=1..45);
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CROSSREFS
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Cf. A106199.
Sequence in context: A044510 A088248 A078111 this_sequence A099865 A065008 A116353
Adjacent sequences: A106197 A106198 A106199 this_sequence A106201 A106202 A106203
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KEYWORD
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frac,nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 24 2005
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