

A278452


a(n) = nearest integer to b(n) = c^(b(n1)/(n1)), where c = e = 2.71828... and b(1) is chosen such that the sequence neither explodes nor goes to 1.


7



1, 4, 7, 11, 15, 19, 23, 28, 33, 37, 42, 48, 53, 58, 64, 69, 75, 80, 86, 92, 97, 103, 109, 115, 121, 127, 133, 139, 146, 152, 158, 165, 171, 177, 184, 190, 197, 203, 210, 216, 223, 230, 236, 243, 250, 256, 263, 270, 277, 284, 290, 297, 304, 311, 318, 325, 332, 339, 346, 353, 360, 367, 375, 382, 389, 396, 403, 410, 418, 425
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OFFSET

1,2


COMMENTS

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n1)b(n+1)/b(n)^2 < 1 (due to rounding to the nearest integer a(n1)a(n+1)/a(n)^2 is not always less than 1).
In this case b(1) = 1.3679012617... A278812. If b(1) were chosen smaller the sequence would approach 1, if it were chosen greater the sequence would at some point violate b(n1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=e similar): "Suppose one starts with b(1) = 2, the sequence would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3."
b(n) = n*log((n+1)*log((n+2)*log(...))) ~ n*log(n).  Andrey Zabolotskiy, Dec 01 2016


LINKS

Rok Cestnik, Table of n, a(n) for n = 1..1000
Rok Cestnik, Plot of the dependence of b(1) on c


EXAMPLE

a(2) = round(e^1.36...) = round(3.92...) = 4.
a(3) = round(e^(3.92.../2)) = round(7.12...) = 7.
a(4) = round(e^(7.12.../3)) = round(10.74...) = 11.


MATHEMATICA

c = E;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
For[d = 0, d < 9, ++d,
b1 = b1 + 1/10^p;
bn = b1;
For[i = 1, i < Round[n*1.2], ++i,
bn = N[c^(bn/i), acc];
If[bn > th, Break[]];
];
If[bn > th, {
b1 = b1  1/10^p;
Break[];
}];
];
];
bnlist = {N[b1]};
bn = b1;
For[i = 1, i < n, ++i,
bn = N[c^(bn/i), acc];
If[bn > th, Break[]];
bnlist = Append[bnlist, N[bn]];
];
anlist = Map[Round[#] &, bnlist]


CROSSREFS

For decimal expansion of b(1) see A278812.
For different values of c see A278448, A278449, A278450, A278451.
For b(1)=0 see A278453.
Sequence in context: A084390 A101741 A247432 * A056548 A065981 A244240
Adjacent sequences: A278449 A278450 A278451 * A278453 A278454 A278455


KEYWORD

nonn


AUTHOR

Rok Cestnik, Nov 22 2016


STATUS

approved



