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A005728
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Number of fractions in Farey series of order n; a(n) = 1 + A002088(n).
(Formerly M0661)
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30
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1, 2, 3, 5, 7, 11, 13, 19, 23, 29, 33, 43, 47, 59, 65, 73, 81, 97, 103, 121, 129, 141, 151, 173, 181, 201, 213, 231, 243, 271, 279, 309, 325, 345, 361, 385, 397, 433, 451, 475, 491, 531, 543, 585, 605, 629, 651, 697, 713, 755, 775, 807, 831, 883, 901, 941, 965
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OFFSET
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0,2
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COMMENTS
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Sometimes called Phi(n).
Leo Moser found an interesting way to generate this sequence, see Gardner.
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REFERENCES
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M. Gardner, The Last Recreations, 1997, chap 12.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, a foundation for computer science, Chapter 4.5 - Relative Primality, pages 118 - 120 and Chapter 9 - Asymptotics, Problem 6, pages 448 - 449, Addison-Wesley Publishing Co., Reading, Mass., 1989.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
S. A. Khan, How Many Equivalent Resistances?, RESONANCE, May 2012; http://www.ias.ac.in/resonance/May2012/p468-475.pdf. - From N. J. A. Sloane, Oct 15 2012
S. A. Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162. - From N. J. A. Sloane, Oct 23 2012
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
Sameen Ahmed KHAN, Mathematica notebook
Eric Weisstein's World of Mathematics, Farey Sequence.
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FORMULA
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a(n) = 1 + Sum_{i=1..n} phi(i).
a(n) = n(n+3)/2 - Sum(k = 2 to n, a([n/k])). - David W. Wilson, May 25, 2002
a(n) = a(n-1) + phi(n) with a(0) = 1. - Arkadiusz Wesolowski, Oct 13 2012
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EXAMPLE
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a(5)=11 because the fractions are 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1.
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MATHEMATICA
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Accumulate@ Array[ EulerPhi, 54, 0] + 1
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PROG
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(Haskell)
a005728 n = a005728_list
a005728_list = scanl (+) 1 a000010_list
-- Reinhard Zumkeller, Aug 04 2012
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CROSSREFS
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Essentially the same as A049643. Cf. A006843, A002088, A055197, A055201.
Sequence in context: A152900 A079151 A049643 * A050437 A096246 A106639
Adjacent sequences: A005725 A005726 A005727 * A005729 A005730 A005731
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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