A278453 -id:A278453 - cp's OEIS frontend

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.

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]

A278448 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

3, 7, 13, 19, 25, 32, 39, 46, 53, 61, 69, 77, 85, 93, 102, 110, 119, 128, 136, 145, 154, 163, 173, 182, 191, 201, 210, 220, 229, 239, 248, 258, 268, 278, 288, 298, 307, 318, 328, 338, 348, 358, 368, 379, 389, 399, 410, 420, 430, 441, 451, 462, 473, 483, 494, 505, 515, 526, 537, 547, 558, 569, 580, 591, 602, 613, 624, 635, 646, 657

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) = 2.8718808270... A278808. 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. 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_2((n+1)*log_2((n+2)*log_2(...))) ~ n*log_2(n). - Andrey Zabolotskiy, Dec 01 2016

			a(2) = round(2^2.87...) = round(7.32...) = 7.
a(3) = round(2^(7.32.../2)) = round(12.64...) = 13.
a(4) = round(2^(12.64.../3)) = round(18.55...) = 19.

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 A278808.
For different values of c see A278449, A278450, A278451, A278452.
For b(1)=0 see A278453.

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[];
      }];
    ];
  ];
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]

A278449 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=3 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

1, 3, 6, 9, 13, 16, 20, 24, 29, 33, 37, 42, 47, 51, 56, 61, 66, 71, 76, 81, 86, 92, 97, 102, 108, 113, 118, 124, 129, 135, 141, 146, 152, 158, 163, 169, 175, 181, 187, 193, 199, 205, 210, 216, 222, 229, 235, 241, 247, 253, 259, 265, 271, 278, 284, 290, 296, 303, 309, 315, 322, 328, 334, 341, 347, 354, 360, 367, 373, 379

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) = 1.0828736095... A278809. 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=3 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_3((n+1)*log_3((n+2)*log_3(...))) ~ n*log_3(n). - Andrey Zabolotskiy, Dec 01 2016

			a(2) = round(3^1.08...) = round(3.28...) = 3.
a(3) = round(3^(3.28.../2)) = round(6.07...) = 6.
a(4) = round(3^(6.07.../3)) = round(9.26...) = 9.

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 A278809.
For different values of c see A278448, A278450, A278451, A278452.
For b(1)=0 see A278453.

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[];
      }];
    ];
  ];
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]

A278450 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=4 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

Original entry on oeis.org

0, 2, 4, 6, 9, 12, 14, 17, 21, 24, 27, 31, 34, 38, 41, 45, 49, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 101, 105, 109, 114, 118, 122, 127, 131, 135, 140, 144, 149, 153, 158, 162, 167, 172, 176, 181, 185, 190, 195, 200, 204, 209, 214, 218, 223, 228, 233, 238, 242, 247, 252, 257, 262, 267, 272, 277, 282, 287

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) = 0.4970450000... A278810. 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=4 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_4((n+1)*log_4((n+2)*log_4(...))) ~ n*log_4(n). - Andrey Zabolotskiy, Dec 01 2016

			a(2) = round(4^0.49...) = round(1.99...) = 2.
a(3) = round(4^(1.99.../2)) = round(3.97...) = 4.
a(4) = round(4^(3.97.../3)) = round(6.28...) = 6.

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 A278810.
For different values of c see A278448, A278449, A278451, A278452.
For b(1)=0 see A278453.

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[];
      }];
    ];
  ];
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]

A278451 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), 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, 3, 5, 7, 9, 11, 14, 17, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 56, 59, 62, 66, 69, 73, 76, 80, 83, 87, 90, 94, 98, 101, 105, 109, 112, 116, 120, 123, 127, 131, 135, 139, 143, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 231, 235, 239

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) = 0.1775819188... A278811. 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=5 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_5((n+1)*log_5((n+2)*log_5(...))) ~ n*log_5(n). - Andrey Zabolotskiy, Dec 01 2016

			a(2) = round(5^0.17...) = round(1.33...) = 1.
a(3) = round(5^(1.33.../2)) = round(2.91...) = 3.
a(4) = round(5^(2.91.../3)) = round(4.78...) = 5.

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 A278811.
For different values of c see A278448, A278449, A278450, A278452.
For b(1)=0 see A278453.

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[];
      }];
    ];
  ];
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]

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.

Original entry on oeis.org

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]

cp's OEIS Frontend

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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A278448 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=2 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

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A278449 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=3 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

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A278450 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=4 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

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A278451 a(n) = nearest integer to b(n) = c^(b(n-1)/(n-1)), where c=5 and b(1) is chosen such that the sequence neither explodes nor goes to 1.

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Mathematica

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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Mathematica