0,2

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.

Comment from Hans Havermann: I calculated these with Mathematica. I used NSum[1/(2i), {i, 1, x}] for the even denominators, where I had to adjust the options to obtain maximal accuracy and N[(EulerGamma + Log[4] - 2)/2 + PolyGamma[0, 3/2 + y]/2, precision] for the odd denominators. The precision needed for the last term shown was around 45 digits.

B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964; see p. 55.

1 - (1/2 + 1/4 + 1/6 + ... + 1/454) = -2.002183354..., which is just less than -2; so a(1) = m_1 = 227.

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(2) = m_2 = 22945.

Cf. A092267 (essentially the same), A002387, A056053, A092318, A092317, A092315.

Cf. A092273.

Sequence in context: A031693 A158228 A115998 this_sequence A122976 A098245 A090943

Adjacent sequences: A092321 A092322 A092323 this_sequence A092325 A092326 A092327

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com), Feb 16 2004

a(2) and a(3) from Hugo Pfoertner (hugo(AT)pfoertner.org), Feb 17 2004

a(4) onwards from Hans Havermann (pxp(AT)rogers.com), Feb 18 2004