

A225922


a(n) is the least k such that f(a(n1)+1) + ... + f(k) > f(a(n2)+1) + ... + f(a(n1)) 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
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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



