A278812 -id:A278812 - cp's OEIS frontend

A278452 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c = e = 2.71828... and b(1) is chosen such that the sequence neither explodes nor goes to 1.

1, 4, 7, 11, 15, 19, 23, 28, 33, 37, 42, 48, 53, 58, 64, 69, 75, 80, 86, 92, 97, 103, 109, 115, 121, 127, 133, 139, 146, 152, 158, 165, 171, 177, 184, 190, 197, 203, 210, 216, 223, 230, 236, 243, 250, 256, 263, 270, 277, 284, 290, 297, 304, 311, 318, 325, 332, 339, 346, 353, 360, 367, 375, 382, 389, 396, 403, 410, 418, 425

Offset: 1

Rok Cestnik, Nov 22 2016

nonn

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1 (due to rounding to the nearest integer a(n-1)a(n+1)/a(n)^2 is not always less than 1).
In this case b(1) = 1.3679012617... A278812. If b(1) were chosen smaller the sequence would approach 1, if it were chosen greater the sequence would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=e similar): "Suppose one starts with b(1) = 2, the sequence would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3."
b(n) = n*log((n+1)*log((n+2)*log(...))) ~ n*log(n). - Andrey Zabolotskiy, Dec 01 2016

			a(2) = round(e^1.36...) = round(3.92...) = 4.
a(3) = round(e^(3.92.../2)) = round(7.12...) = 7.
a(4) = round(e^(7.12.../3)) = round(10.74...) = 11.

Rok Cestnik, Table of n, a(n) for n = 1..1000
Rok Cestnik, Plot of the dependence of b(1) on c

For decimal expansion of b(1) see A278812.
For different values of c see A278448, A278449, A278450, A278451.
For b(1)=0 see A278453.

c = E;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
bnlist = {N[b1]};
bn = b1;
For[i = 1, i < n, ++i,
  bn = N[c^(bn/i), acc];
  If[bn > th, Break[]];
  bnlist = Append[bnlist, N[bn]];
  ];
anlist = Map[Round[#] &, bnlist]

A278808 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278448, where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

2, 8, 7, 1, 8, 8, 0, 8, 2, 7, 0, 4, 5, 4, 5, 4, 6, 5, 8, 8, 9, 0, 5, 5, 1, 7, 5, 5, 0, 4, 5, 7, 5, 0, 4, 5, 8, 6, 5, 6, 5, 2, 5, 1, 1, 8, 4, 7, 9, 6, 5, 6, 5, 6, 7, 9, 2, 9, 9, 5, 4, 0, 1, 0, 8, 4, 0, 4, 5, 7, 9, 6, 8, 3, 0, 8, 9, 2, 7, 0, 3, 6, 0, 1, 8, 2, 8, 6, 3, 8, 1, 8, 6, 7, 6, 7, 8, 7, 5, 4, 8, 0, 8, 4, 3

Offset: 1

Rok Cestnik, Nov 28 2016

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.
If b(1) were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Suppose one starts with b(1) = 2, the sequence b(n) would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3.
No closed form expression is known. Probably transcendental but this is unproved. - Robert G. Wilson v, Dec 01 2016

			2.87188082704545465889055175504575045865652511847965...

Rok Cestnik, Table of n, a(n) for n = 1..1000
Rok Cestnik, Plot of the dependence of b(1) on c

For sequence round(b(n)) see A278448.
For different values of c see A278809, A278810, A278811, A278812.
For b(1)=0 see A278813.

c = 2;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
N[b1,n]
RealDigits[Fold[Log2[#1*#2] &, 1, Reverse@Range[2, 144]], 10,
111][[1]] (* Robert G. Wilson v, Dec 01 2016 *)

log_2(2*log_2(3*log_2(4*log_2(...)))). - Andrey Zabolotskiy, Nov 30 2016

A278809 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278449, where c=3 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

1, 0, 8, 2, 8, 7, 3, 6, 0, 9, 5, 2, 0, 7, 3, 8, 6, 9, 4, 0, 8, 2, 8, 5, 0, 3, 1, 3, 4, 5, 3, 1, 0, 0, 8, 0, 2, 5, 7, 8, 6, 3, 4, 5, 4, 7, 8, 5, 3, 8, 5, 0, 6, 4, 3, 2, 8, 8, 4, 7, 8, 2, 1, 6, 8, 0, 6, 9, 2, 2, 7, 8, 8, 9, 5, 2, 9, 9, 5, 5, 7, 4, 7, 0, 6, 8, 1, 4, 4, 8, 7, 8, 6, 2, 3, 9, 2, 4, 4, 3, 1, 1, 5, 4, 5, 9, 9, 1, 8, 9, 2, 4, 3, 8, 8, 4, 0, 6, 3, 6, 2, 6, 1, 3, 5, 9, 3, 4, 0, 0

Offset: 1

Rok Cestnik, Nov 28 2016

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.
If b(1) were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=3 similar): "Suppose one starts with b(1) = 2, the sequence b(n) would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3."

			1.08287360952073869408285031345310080257863454785385...

Rok Cestnik, Table of n, a(n) for n = 1..1000
Rok Cestnik, Plot of the dependence of b(1) on c

For sequence round(b(n)) see A278449.
For different values of c see A278808, A278810, A278811, A278812.
For b(1)=0 see A278813.

c = 3;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
N[b1,n]
RealDigits[ Fold[ Log[3, #1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* Robert G. Wilson v, Dec 02 2016 *)

log_3(2*log_3(3*log_3(4*log_3(...)))). - Andrey Zabolotskiy, Dec 01 2016

A278810 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278450, where c=4 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

0, 4, 9, 7, 0, 4, 5, 0, 0, 0, 0, 7, 5, 8, 9, 4, 5, 0, 7, 7, 3, 7, 8, 3, 7, 6, 1, 5, 5, 2, 9, 6, 6, 8, 9, 3, 6, 1, 4, 2, 3, 9, 3, 2, 4, 7, 9, 8, 5, 9, 3, 8, 9, 5, 9, 8, 3, 0, 3, 6, 8, 4, 6, 1, 2, 7, 6, 0, 5, 6, 6, 4, 4, 3, 1, 8, 7, 3, 5, 5, 7, 9, 7, 8, 8, 3, 6, 3, 2, 4, 9, 8, 4, 6, 7, 7, 2, 1, 6, 2, 5, 2, 9, 5, 7, 5, 7, 6, 5, 3, 0, 8, 0, 1, 4, 5, 3, 8, 6, 4, 1, 6, 3, 9, 7, 6, 9, 8, 9, 3

Offset: 1

Rok Cestnik, Nov 28 2016

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.
If b(1) were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=4 similar): "Suppose one starts with b(1) = 2, the sequence b(n) would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3."

			0.49704500007589450773783761552966893614239324798593...

Rok Cestnik, Table of n, a(n) for n = 1..1000
Rok Cestnik, Plot of the dependence of b(1) on c

For sequence round(b(n)) see A278450.
For different values of c see A278808, A278809, A278811, A278812.
For b(1)=0 see A278813.

c = 4;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
N[b1,n]
RealDigits[ Fold[ Log[4, #1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* Robert G. Wilson v, Dec 02 2016 *)

log4(2*log4(3*log4(4*log4(...)))). - Andrey Zabolotskiy, Nov 30 2016

A278811 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278451, where c=5 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

0, 1, 7, 7, 5, 8, 1, 9, 1, 8, 8, 0, 2, 5, 1, 4, 0, 3, 3, 3, 8, 3, 5, 0, 3, 1, 8, 1, 3, 0, 8, 6, 6, 9, 8, 5, 7, 8, 8, 3, 2, 9, 7, 7, 0, 3, 4, 6, 8, 1, 0, 5, 2, 1, 5, 6, 4, 2, 3, 6, 3, 5, 7, 4, 3, 3, 3, 1, 7, 4, 8, 3, 6, 8, 4, 2, 2, 1, 1, 8, 3, 5, 1, 4, 8, 4, 6, 9, 0, 7, 6, 9, 7, 1, 4, 2, 7, 2, 6, 5, 7, 5, 1, 5, 6, 9, 2, 7, 7, 0, 1, 6, 5, 4, 1, 3, 4, 9, 9, 8, 6, 1, 3, 5, 5, 3, 1, 5, 8, 5

Offset: 1

Rok Cestnik, Nov 28 2016

For the given c there exists a unique b(1) for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.
If b(1) were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of b(1) is found through trial and error. Illustrative example for the case of c=2 (for c=5 similar): "Suppose one starts with b(1) = 2, the sequence b(n) would continue b(2) = 4, b(3) = 4, b(4) = 2.51..., b(5) = 1.54... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say b(1) = 3, which gives rise to b(2) = 8, b(3) = 16, b(4) = 40.31... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of b(1) is between 2 and 3."

			0.17758191880251403338350318130866985788329770346810...

Rok Cestnik, Table of n, a(n) for n = 1..1000
Rok Cestnik, Plot of the dependence of b(1) on c

For sequence round(b(n)) see A278451.
For different values of c see A278808, A278809, A278810, A278812.
For b(1)=0 see A278813.

c = 5;
n = 100;
acc = Round[n*1.2];
th = 1000000;
b1 = 0;
For[p = 0, p < acc, ++p,
  For[d = 0, d < 9, ++d,
    b1 = b1 + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i,
     bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];
     ];
    If[bn > th, {
      b1 = b1 - 1/10^p;
      Break[];
      }];
    ];
  ];
N[b1,n]
RealDigits[ Fold[ Log[5, #1*#2] &, 1, Reverse@ Range[2, 160]], 10, 111][[1]] (* Robert G. Wilson v, Dec 02 2016 *)

log5(2*log5(3*log5(4*log5(...)))). - Andrey Zabolotskiy, Nov 30 2016

A278813 Decimal expansion of c in the sequence b(n+1) = c^(b(n)/n) A278453, where b(1)=0 and c is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

5, 7, 5, 8, 1, 9, 5, 9, 3, 9, 1, 1, 0, 3, 7, 4, 9, 4, 1, 9, 7, 4, 0, 2, 8, 8, 6, 5, 0, 0, 9, 3, 2, 9, 0, 9, 2, 4, 7, 4, 2, 4, 2, 6, 4, 7, 0, 5, 5, 3, 1, 5, 4, 1, 5, 1, 4, 1, 2, 5, 9, 9, 0, 6, 1, 9, 7, 1, 0, 7, 5, 9, 8, 9, 1, 5, 8, 7, 2, 3, 0, 8, 3, 3, 3, 7, 8, 7, 0, 6, 9, 5, 8, 7, 9, 1, 1, 5, 7, 2, 0, 0, 5, 6, 2, 9, 5, 0, 5, 6, 3, 2, 1, 1, 0, 5, 7, 1, 4, 7, 1, 3, 5, 9, 5, 0, 6, 0, 7, 7

Offset: 1

Rok Cestnik, Nov 28 2016

There exists a unique value of c for which the sequence b(n) does not converge to 1 and at the same time always satisfies b(n-1)b(n+1)/b(n)^2 < 1.
If c were chosen smaller the sequence b(n) would approach 1, if it were chosen greater it would at some point violate b(n-1)b(n+1)/b(n)^2 < 1 and from there on quickly escalate.
The value of c is found through trial and error. Suppose one starts with c = 5, the sequence b(n) would continue b(2) = 1, b(3) = 2.23..., b(4) = 3.31..., b(5) = 3.80..., b(6) = 3.39..., b(7) = 2.48..., b(8) = 1.77... and from there one can see that such a sequence is tending to 1. One continues by trying a larger value, say c = 6, which gives rise to b(2) = 1, b(3) = 2.44, b(4) = 4.31..., b(5) = 6.92..., b(6) = 11.94..., b(7) = 35.38... and from there one can see that such a sequence is escalating too fast. Therefore, one now knows that the true value of c is between 5 and 6.
c satisfies 2*log_c(3*log_c(4*log_c(...))) = 1. - Andrey Zabolotskiy, Dec 02 2016
No closed form expression is known. Probably transcendental but this is unproved. - Robert G. Wilson v, Dec 02 2016

			5.75819593911037494197402886500932909247424264705531...

Robert G. Wilson v, Table of n, a(n) for n = 1..2500 (first 1000 from Rok Cestnik)
Rok Cestnik, Plot of the dependence of b(1) on c

For sequence round(b(n)) see A278453.

For different values of b(1) see A278808, A278809, A278810, A278811, A278812.

b1 = 0;
n = 100;
acc = Round[n*1.2];
th = 1000000;
c = 0;
For[p = 0, p < acc, ++p, For[d = 0, d < 9, ++d, c = c + 1/10^p;
    bn = b1;
    For[i = 1, i < Round[n*1.2], ++i, bn = N[c^(bn/i), acc];
     If[bn > th, Break[]];];
    If[bn > th, {c = c - 1/10^p;
      Break[];
      }];
    ];
  ];
N[c, n]

A277313 Decimal expansion of the nested logarithm log(1+log(2+log(3+log(4+...)))).

Original entry on oeis.org

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

Alex Klotz, Oct 09 2016

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

			0.82035986220878978847346679494...

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

Similar in concept to A072449.

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

// 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 */

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

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 *)

More digits from Alois P. Heinz, Oct 09 2016

A352495 Decimal expansion of the pearl of the Riemann zeta function.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 7, 8, 5, 7, 6, 3, 3, 0, 6, 6, 4, 4, 0, 7, 3, 0, 2, 1, 5, 0, 9, 1, 8, 5, 7, 3, 6, 2, 1, 7, 7, 8, 2, 9, 7, 1, 0, 0, 9, 1, 4, 0, 5, 3, 3, 3, 0, 4, 7, 8, 7, 9, 7, 3, 1, 9, 2, 8, 4, 5, 8, 6, 4, 7, 3, 5, 4, 1, 6, 6, 6, 1, 2, 9, 3, 5, 2, 6, 5, 0, 0

Offset: 1

Eduard Roure Perdices, Mar 18 2022

Let Z be the Riemann zeta function, and consider its sequence of nontrivial zeros with nonnegative imaginary part, {r(m)}, so that for every m >= 1, Z(r(m)) = 0, 0 <= Re(r(m)) <= 1, and 0 <= Im(r(m)), and for every k > m, Im(r(m)) < Im(r(k)), or Im(r(m)) = Im(r(k)) and Re(r(m)) < Re(r(k)).
Let i be the imaginary unit, and define the sequence {b(m)} as follows: b(1) = Z((r(1)-1/2)/i), b(2) = Z((r(1)-1/2)/i + Z((r(2)-1/2)/i)), b(3) = Z((r(1)-1/2)/i + Z((r(2)-1/2)/i + Z((r(3)-1/2)/i))), and so on. If this sequence converges, we call its limit the pearl of Z.
Suppose that the Riemann Hypothesis is true. Then the sequence {b(m)} is real. On the interval [2,oo), Z is decreasing, positive, and bounded above by 2, so {b(2*m-1)} is decreasing and bounded below by 0, and hence, it converges to a real value, say A. Moreover, {b(2*m)} is increasing and b(2*m) <= b(2*m+1), and by repeated application of the mean value theorem, b(2*m+1) - b(2*m) <= Z(Im(r(2*m+1))) * |Z'(Im(r(1)))|^(2*m) <= 2*(4/100000)^(2*m), so {b(2*m)} also converges to A, and {a(n)} is the decimal expansion of this value.
We don't know if the existence of a real pearl of Z implies the Riemann Hypothesis.
More generally, the definition of pearl works for Dirichlet L-functions, giving rise to analogous constants, not necessarily real.

			1.00002785763306644073021509185736217782971009140533304787973192845864...

Eduard Roure Perdices, Table of n, a(n) for n = 1..5000

Cf. A072449, A099874, A099876, A099877, A105546, A105817, A151558, A239349, A277313, A278812.

RealDigits[Re[res = Fold[Zeta[#1 + #2] &, 0, Reverse[(ZetaZero[Range[10]] - 1/2)/I]]], 10, 100][[1]]

A285814 Decimal expansion of the limit of the nested logarithm log(1+2log(1+3log(1+4*log(...)))).

Original entry on oeis.org

1, 6, 6, 3, 0, 2, 7, 2, 6, 4, 5, 9, 5, 6, 8, 9, 5, 2, 1, 1, 3, 2, 6, 6, 7, 6, 6, 2, 6, 8, 4, 6, 8, 4, 1, 8, 8, 8, 9, 3, 0, 9, 2, 9, 8, 0, 6, 3, 8, 8, 1, 9, 1, 0, 3, 3, 1, 8, 3, 2, 4, 3, 3, 1, 9, 6, 2, 7, 0, 1, 9, 6, 5, 6, 4, 1, 4, 1, 3, 5, 1, 1, 3, 6, 5, 7, 6, 4, 9, 7, 0, 6, 9, 7, 1, 2, 4, 4, 9, 2, 0, 4, 6, 0, 6

Offset: 1

Alex Klotz and Robert G. Wilson v, Apr 27 2017

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

			1.6630272645956895211326676626846841888930929806388191033183243319627019656...

Alex Klotz, The Nested Logarithm Constants

Cf. A072449, A276521, A276538, A276539, A277313, A278812, A283749.

bgn = 147; RealDigits[ Fold[ N[1 + #2*Log@#1, 200] &, bgn +1, Reverse@Range@bgn] -1, 10, 111][[1]]

cp's OEIS Frontend

A278452 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c = e = 2.71828... and b(1) is chosen such that the sequence neither explodes nor goes to 1.

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A278808 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278448, where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

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A278809 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278449, where c=3 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

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A278810 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278450, where c=4 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

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A278811 Decimal expansion of b(1) in the sequence b(n+1) = c^(b(n)/n) A278451, where c=5 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

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A278813 Decimal expansion of c in the sequence b(n+1) = c^(b(n)/n) A278453, where b(1)=0 and c is chosen such that the sequence neither explodes nor goes to 1.

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A277313 Decimal expansion of the nested logarithm log(1+log(2+log(3+log(4+...)))).

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C

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A352495 Decimal expansion of the pearl of the Riemann zeta function.

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A285814 Decimal expansion of the limit of the nested logarithm log(1+2log(1+3log(1+4*log(...)))).