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A092267
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Values 2m_0+1 = 1, 2m_1, 2m_2+1, ... associated with divergent series T shown below.
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2
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1, 454, 45891, 547208496, 3013267310449, 1961694770407970734, 589785633779065944213245, 20963601300674244910397534828794, 344117353602393170461608383214200982125
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| T = 1
- (1/2 + 1/4 + 1/6 + ... + 1/(2m_1))
+ (1/3 + 1/5 + 1/7 + ... + 1/(2m_2+1))
- (1/(2m_1+2) + 1/(2m_1+4) + ... + 1/(2m_3)
+ (1/(2m_2+3) + 1/(2m_2+5) + ... + 1/(2m_4+1))
- (1/(2m_3+2) + 1/(2m_3+4) + ... + 1/(2m_5)
+ (1/(2m_4+3) + 1/(2m_4+5) + ... + 1/(2m_6+1))
- ...
where the partial sums of the terms from 1 through the end of rows 0, 1, ... are respectively 1, just < -2, just > 3, just < -4, just > 5, etc.
Every positive number appears exactly once as a denominator in T.
The series T is a divergent rearrangement of the conditionally convergent series Sum_{ j>=1} (-1)^j/j which has the entire real number system as its set of limit points.
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REFERENCES
| B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964; see p. 55.
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EXAMPLE
| 1 - (1/2 + 1/4 + 1/6 + ... + 1/454) = -2.002183354..., which is just less than -2; so a(1) = 2m_1 = 454.
1 - (1/2 + 1/4 + 1/6 + ... + 1/454) + (1/3 + 1/5 + ... + 1/45891) = 3.000021113057..., which is just greater than 3; so a(1) = 2m_2 + 1 = 45891.
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CROSSREFS
| Cf. A092324 (essentially the same), A002387, A056053, A092318, A092317, A092315.
Cf. A092273.
Sequence in context: A206217 A081739 A076547 * A206338 A123563 A043475
Adjacent sequences: A092264 A092265 A092266 * A092268 A092269 A092270
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Feb 16 2004
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EXTENSIONS
| a(2) and a(3) from Hugo Pfoertner (hugo(AT)pfoertner.org), Feb 17 2004
a(4) onwards from Hans Havermann (gladhobo(AT)teksavvy.com), Feb 18 2004
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