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A225922 a(n) is the least k such that f(a(n-1)+1) + ... + f(k) > f(a(n-2)+1) + ... + f(a(n-1)) for n > 1, where f(n) = 1/(n+7) and a(1) = 1. 1
1, 16, 58, 176, 507, 1436, 4043, 11359, 31890, 89506, 251193, 704933, 1978258, 5551574, 15579326, 43720081, 122691130, 344306598, 966223316 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: a(n) is linearly recurrent. See A225918 for details.

LINKS

Table of n, a(n) for n=1..19.

EXAMPLE

a(1) = 1 by decree; a(2) = 15 because 1/9 + ... + 1/21 < 1 < 1/9 + ... + 1/(15+7), so that a(3) = 58 because 1/23 + ... + 1/57 < 1/9 + ... + 1/22 < 1/23 + ... + 1/(58+7).

Successive values of a(n) yield a chain: 1 < 1/(1+8) + ... + 1/(15+7) < 1/(15+8) + ... + 1/(58+7) < 1/(58+8) + ... + 1/(176+7) < ...

Abbreviating this chain as b(1) = 1 < b(2) < b(3) < b(4) < ... < R = 2.80628..., it appears that lim_{n->infinity} b(n) = log(R) = 1.03186... .

MATHEMATICA

nn = 11; f[n_] := 1/(n + 7); a[1] = 1; g[n_] := g[n] = Sum[f[k], {k, 1, n}]; s = 0; a[2] = NestWhile[# + 1 &, 2, ! (s += f[#]) >= a[1] &]; s = 0; a[3] = NestWhile[# + 1 &, a[2] + 1, ! (s += f[#]) >= g[a[2]] - f[1] &]; Do[s = 0; a[z] = NestWhile[# + 1 &, a[z - 1] + 1, ! (s += f[#]) >= g[a[z - 1]] - g[a[z - 2]] &], {z, 4, nn}]; m = Map[a, Range[nn]]

CROSSREFS

Cf. A225918.

Sequence in context: A253428 A005905 A177890 * A235510 A220974 A063521

Adjacent sequences:  A225919 A225920 A225921 * A225923 A225924 A225925

KEYWORD

nonn,more

AUTHOR

Clark Kimberling, May 21 2013

EXTENSIONS

a(10)-a(17) from Robert G. Wilson v, May 22 2013

a(18) from Robert G. Wilson v, Jun 13 2013

a(19) from Jinyuan Wang, Jun 14 2020

STATUS

approved

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Last modified December 5 06:30 EST 2021. Contains 349532 sequences. (Running on oeis4.)