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A277313 Decimal expansion of the nested logarithm log(1+log(2+log(3+log(4+...)))). 3
8, 2, 0, 3, 5, 9, 8, 6, 2, 2, 0, 8, 7, 8, 9, 7, 8, 8, 4, 7, 3, 4, 6, 6, 7, 9, 4, 9, 4, 0, 6, 3, 9, 1, 5, 8, 4, 1, 5, 9, 0, 9, 7, 5, 3, 4, 1, 3, 1, 6, 1, 9, 3, 7, 6, 5, 4, 6, 8, 7, 6, 7, 4, 9, 4, 8, 5, 0, 2, 4, 0, 7, 0, 1, 9, 2, 2, 9, 3, 8, 4, 6, 3, 2, 4, 5, 1, 7, 7, 4, 5, 4, 4, 7, 9, 2, 9, 9, 2, 8, 8, 2, 9, 8, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Found empirically. Logarithms are natural.

Converges to within 10^-4 of the asymptotic value when the innermost term is 7. First fifteen decimals can be found numerically by using 17 nested terms.

No closed form expression is known. Probably transcendental but this is unproved.

Empirically, the number of bits of precision with N as the innermost term is 0.02N^2 + 2.24N - 8.5. This means that using N as the largest innermost term gives (0.02N^2 + 2.24N - 8.5)*(log_10(2)) digits. - Cade Brown, Oct 10 2016

LINKS

Cade Brown, Table of n, a(n) for n = 0..14997

EXAMPLE

0.82035986220878978847346679494...

MATHEMATICA

RealDigits[SequenceLimit[N[Table[Log[Fold[#2 + Log[#1] &, Reverse@Range[n]]], {n, 1, 100}], 200]], 10, 105][[1]] (* Vladimir Reshetnikov, Oct 11 2016 *)

RealDigits[ Fold[ Log[#1 + #2] &, 0, Reverse[ Range[74]]], 10, 111][[1]] (* Robert G. Wilson v, Oct 26 2016 *)

PROG

(MATLAB)

x=100;

for i=99:-1:1

x=log(i+x);

end

%the initial value of x can be increased for greater precision, but it converges starting well below 100

(C)

// Computes b bits, and uses MPFR for multiprecision.

#include <mpfr.h>

#include <stdio.h>

#include <math.h>

int main() {

    int b=256, i;

    int N = 500 + (int)(4 * floor(-56+sqrt(3561+50*b)));

    mpfr_t m;

    mpfr_init2(m, b);

    mpfr_set_ui(m, N, rnd);

    for (i = N; i > 0; --i) {

        mpfr_log(m, m, MPFR_RNDN);

        mpfr_add_ui(m, m, i - 1, MPFR_RNDN);

    }

    mpfr_printf("\nval %.*Rf\n\n", b - 10, m);

    mpfr_clear(m);

} /* Cade Brown, Oct 10 2016 */

CROSSREFS

Similar in concept to A072449.

Cf. A278812 (log(2*log(3*log(4*...))), (or log(2) + log(log(3) + log(log(4) + ...))))).

Sequence in context: A209244 A021850 A197576 * A182170 A011105 A098829

Adjacent sequences:  A277310 A277311 A277312 * A277314 A277315 A277316

KEYWORD

nonn,cons

AUTHOR

Alex Klotz, Oct 09 2016

EXTENSIONS

More digits from Alois P. Heinz, Oct 09 2016

STATUS

approved

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Last modified May 24 09:23 EDT 2017. Contains 286963 sequences.