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A206911 Position of n-th partial sum of the harmonic series when all the partial sums are jointly ranked with the set {log(k+1)}; complement of A206912. 7
2, 5, 8, 11, 13, 16, 19, 22, 24, 27, 30, 33, 36, 38, 41, 44, 47, 49, 52, 55, 58, 61, 63, 66, 69, 72, 74, 77, 80, 83, 86, 88, 91, 94, 97, 100, 102, 105, 108, 111, 113, 116, 119, 122, 125, 127, 130, 133, 136, 138, 141, 142, 143, 144, 145, 146, 147, 148, 149 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Conjecture: the difference sequence of A206911 consists of 2s and 3s, and the ratio (number of 3s)/(number of 2s) tends to a number between 3.5 and 3.6.
Similar conjectures can be stated for difference sequences based on jointly ranked sets, such as A206903, A206906, A206928, A206805, A206812, and A206815.
LINKS
EXAMPLE
Let S(n)=1+1/2+1/3+...+1/n and L(n)=log(n+1). Then
L(1)<S(1)<L(2)<L(3)<S(2)<L(4)<L(5)<S(3)<L(6)<..., so that
A206911=(2,5,8,...).
MATHEMATICA
f[n_] := Sum[1/k, {k, 1, n}]; z = 300;
g[n_] := N[Log[n + 1]];
c = Table[f[n], {n, 1, z}];
s = Table[g[n], {n, 1, z}];
j = Sort[Union[c, s]];
p[n_] := Position[j, f[n]]; q[n_] := Position[j, g[n]];
Flatten[Table[p[n], {n, 1, z}]] (* A206911 *)
Flatten[Table[q[n], {n, 1, z}]] (* A206912 *)
CROSSREFS
Sequence in context: A187341 A329924 A330112 * A093609 A249118 A292643
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 13 2012
STATUS
approved

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)