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A277313
Decimal expansion of the nested logarithm log(1+log(2+log(3+log(4+...)))).
4
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
OFFSET
0,1
COMMENTS
Found empirically. Logarithms are natural.
Converges to within 10^-4 of the asymptotic value when the innermost term is 7. The first fifteen digits after the decimal point 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
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: A338851 A021850 A197576 * A344362 A182170 A011105
KEYWORD
nonn,cons
AUTHOR
Alex Klotz, Oct 09 2016
EXTENSIONS
More digits from Alois P. Heinz, Oct 09 2016
STATUS
approved