A277313 Decimal expansion of the nested logarithm log(1+log(2+log(3+log(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

Author

Alex Klotz, Oct 09 2016

Keywords

• nonn
• cons

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

Examples

			0.82035986220878978847346679494...
		

Links

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

Crossrefs

Similar in concept to A072449.

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

Programs

C

// Computes b bits, and uses MPFR for multiprecision.
#include 
#include 
#include 
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 */

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

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 *)

Extensions

More digits from Alois P. Heinz, Oct 09 2016