a sequence from analysis

[N. J. A. Sloane]

The following are a very nice pair of sequences
for which i could use more terms!

Thanks

Neil

%I A088403
%S A088403 0,227
%N A088403 Values m_0 = 0, m_1, m_2, ... associated with series T shown below.
%C A088403 T = 1
%C A088403 - (1/2 + 1/4 + 1/6 + ... + 1/(2m_1))
%C A088403 + (1/3 + 1/5 + 1/7 + ... + 1/(2m_2+1))
%C A088403 - (1/(2m_1+2) + 1/(2m_1+4) + ... + 1/(2m_3)
%C A088403 + (1/(2m_2+3) + 1/(2m_2+5) + ... + 1/(2m_4+1))
%C A088403 - (1/(2m_3+2) + 1/(2m_3+4) + ... + 1/(2m_5)
%C A088403 + (1/(2m_4+3) + 1/(2m_4+5) + ... + 1/(2m_6+1))
%C A088403 - ...
%C A088403 so that the partial sums of the terms through the ends of the rows are respectively 1, just < -2, just > 3, just < -4, just > 5, etc.
%C A088403 Every positive number appears exactly once as a denominator in T.
%C A088403 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.
%D A088403 B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964; see p. 55.
%e A088403 1 - (1/2 + 1/4 + 1/6 + ... + 1/454) = -2.002183354..., which is just less than -2; so a(1) = m_1 = 227.
%O A088403 0,2
%K A088403 nonn,nice,more,bref
%Y A088403 Cf. A088507.
%A A088403 njas, Feb 16 2004

%I A088507
%S A088507 1,454
%N A088507 Values 2m_0+1 = 1, 2m_1, 2m_2+1, ... associated with series T shown below.
%C A088507 T = 1
%C A088507 - (1/2 + 1/4 + 1/6 + ... + 1/(2m_1))
%C A088507 + (1/3 + 1/5 + 1/7 + ... + 1/(2m_2+1))
%C A088507 - (1/(2m_1+2) + 1/(2m_1+4) + ... + 1/(2m_3)
%C A088507 + (1/(2m_2+3) + 1/(2m_2+5) + ... + 1/(2m_4+1))
%C A088507 - (1/(2m_3+2) + 1/(2m_3+4) + ... + 1/(2m_5)
%C A088507 + (1/(2m_4+3) + 1/(2m_4+5) + ... + 1/(2m_6+1))
%C A088507 - ...
%C A088507 so that the partial sums of the terms through the ends of the rows are respectively 1, just < -2, just > 3, just < -4, just > 5, etc.
%C A088507 Every positive number appears exactly once as a denominator in T.
%C A088507 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.
%D A088507 B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964; see p. 55.
%e A088507 1 - (1/2 + 1/4 + 1/6 + ... + 1/454) = -2.002183354..., which is just less than -2; so a(1) = m_1 = 454.
%O A088507 0,2
%K A088507 nonn,nice,more,bref
%Y A088507 Cf. A088403.
%A A088507 njas, Feb 16 2004


Related sequences: A002387, A056053, A078914, A083027, A084387