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Search: id:A111879
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| A111879 |
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Numerators of array which counts positive rational numbers (not including natural numbers). |
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+0
4
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| 1, 1, 1, 2, 3, 1, 1, 2, 3, 4, 5, 1, 3, 5, 1, 2, 4, 5, 7, 1, 3, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 5, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 3, 5, 9, 11, 1, 2, 4, 7, 8, 11, 13, 1, 3, 5, 7, 9, 11, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 5, 7, 11, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
(list; graph; listen)
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OFFSET
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3,4
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COMMENT
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Denominators are given by A111880.
The sequence of row lengths is [1, 1, 3, 1, 5, 3, 5, 3, 9, 3, 11, 5, 7, 7, ...] = A000010(n)-1 = phi(n)-1, with Euler's totient function phi(n).
For n>=3 delete from the list [seq(j/n-j,j=1..n-2)] the natural numbers and the ratios p/q with (p,q) not 1 (p and q not relatively prime, i.e., p and q have a common divisor >1).
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REFERENCES
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P. Dienes, The Taylor Series, Dover 1957, p. 8, eq.(1).
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LINKS
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W. Lang, Array of ratios and more.
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FORMULA
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a(n, k)=numerator(r(n, k)), n>=3, k=1..phi(n)-1, with phi(n):=A000010(n) (Euler's totient function) and the ratios r(n, k) defined for row n above.
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EXAMPLE
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[1], [1], [1, 2, 3], [1], [1, 2, 3, 4, 5], [1, 3, 5], [1, 2, 4, 5,
7], [1, 3, 7],...
The corresponding ratios are: [1/2], [1/3], [1/4, 2/3, 3/2], [1/5],
[1/6, 2/5, 3/4, 4/3, 5/2], [1/7, 3/5, 5/3], [1/8, 2/7, 4/5, 5/4, 7/2], [1/9,
3/7, 7/3],...
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CROSSREFS
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Row sums give A111881(n)/A069220(n), n>=3, see the W. Lang link.
Cf. A020652/A020653 if natural numbers are included.
Sequence in context: A065882 A007884 A157813 this_sequence A114732 A123338 A152735
Adjacent sequences: A111876 A111877 A111878 this_sequence A111880 A111881 A111882
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KEYWORD
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nonn,easy,frac,tabf
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 23 2005
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