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%I #39 Oct 08 2022 00:01:32
%S 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,
%T 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,
%U 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
%N Decimal expansion of the nested logarithm log(1+log(2+log(3+log(4+...)))).
%C Found empirically. Logarithms are natural.
%C 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.
%C No closed form expression is known. Probably transcendental but this is unproved.
%C 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
%H Cade Brown, <a href="/A277313/b277313.txt">Table of n, a(n) for n = 0..14997</a>
%e 0.82035986220878978847346679494...
%t RealDigits[SequenceLimit[N[Table[Log[Fold[#2 + Log[#1] &, Reverse@Range[n]]], {n, 1, 100}], 200]], 10, 105][[1]] (* _Vladimir Reshetnikov_, Oct 11 2016 *)
%t RealDigits[ Fold[ Log[#1 + #2] &, 0, Reverse[ Range[74]]], 10, 111][[1]] (* _Robert G. Wilson v_, Oct 26 2016 *)
%o (MATLAB)
%o x=100;
%o for i=99:-1:1
%o x=log(i+x);
%o end
%o %the initial value of x can be increased for greater precision, but it converges starting well below 100
%o (C)
%o // Computes b bits, and uses MPFR for multiprecision.
%o #include <mpfr.h>
%o #include <stdio.h>
%o #include <math.h>
%o int main() {
%o int b=256, i;
%o int N = 500 + (int)(4 * floor(-56+sqrt(3561+50*b)));
%o mpfr_t m;
%o mpfr_init2(m, b);
%o mpfr_set_ui(m, N, rnd);
%o for (i = N; i > 0; --i) {
%o mpfr_log(m, m, MPFR_RNDN);
%o mpfr_add_ui(m, m, i - 1, MPFR_RNDN);
%o }
%o mpfr_printf("\nval %.*Rf\n\n", b - 10, m);
%o mpfr_clear(m);
%o } /* _Cade Brown_, Oct 10 2016 */
%Y Similar in concept to A072449.
%Y Cf. A278812 (log(2*log(3*log(4*...))), (or log(2) + log(log(3) + log(log(4) + ...))))).
%K nonn,cons
%O 0,1
%A _Alex Klotz_, Oct 09 2016
%E More digits from _Alois P. Heinz_, Oct 09 2016
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