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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 #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 */ 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 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 July 29 06:21 EDT 2021. Contains 346340 sequences. (Running on oeis4.)