A350133 Last denominator in each run of odd terms in the greedy rearrangement of the alternating harmonic series that converges to 2.

15, 41, 69, 95, 123, 151, 177, 205, 233, 259, 287, 315, 341, 369, 395, 423, 451, 477, 505, 533, 559, 587, 615, 641, 669, 697, 723, 751, 779, 805, 833, 859, 887, 915, 941, 969, 997, 1023, 1051, 1079, 1105, 1133, 1161, 1187, 1215, 1243, 1269, 1297, 1325

Offset

1,1

Comments

The Riemann series theorem tells us that we can rearrange the terms of the alternating harmonic series 1 - 1/2 + 1/3 - 1/4 ... to converge to any number. If we do so using a greedy algorithm to converge to 2, we start with the positive terms 1 + 1/3 + 1/5 + ... + 1/15 which is just over 2, then add the first negative term -1/2, then add 1/17 + 1/19 + ... + 1/41 which is just over 2, then add -1/4, and so on. We take the last denominator in each run as our sequence, so a(1) = 15, a(2) = 41, and so on.

It appears that each subsequent term is either 26 or 28 more than the previous.

Links

Table of n, a(n) for n=1..49.

Wikipedia, Riemann series theorem

Example

For n=3 the third run of terms with odd denominators is 1/43 + 1/45 + ... + 1/69 and so a(3) = 69.

Mathematica

Module[{A = {}, S = 0, k = 1},
Do[While[S < 2, S += 1/k; k += 2]; AppendTo[A, k - 2];
  S -= 1/(2 n), {n, 1, 30}]; A]

Crossrefs

Sequence in context: A219378 A280045 A103003 * A239325 A024845 A056698

Adjacent sequences: A350130 A350131 A350132 * A350134 A350135 A350136

Keyword

nonn

Author

Greg Dresden, Jan 07 2022

Status

approved