A079368 -id:A079368 - cp's OEIS frontend

A079366 Costé prime expansion of Pi - 3.

11, 2, 11, 5, 5, 2, 5, 3, 17, 11, 3, 3, 11, 3, 3, 11, 5, 3, 23, 7, 5, 97, 29, 37, 107, 127, 29, 17, 409, 127, 11, 29, 5, 67, 19, 43, 31, 19, 103, 59, 29, 7, 3, 11, 11, 5, 47, 29, 11, 3, 5, 5, 3, 17, 5, 29, 11, 3, 3, 3, 3, 5, 5, 61, 151, 58889, 1877, 983, 757, 163

Offset: 0

N. J. A. Sloane, Feb 15 2003

For x in (0,1], define P(x) = min{p: p prime, 1/x < p}, Phi(x) = P(x)x - 1. Costé prime expansion of x(0) is sequence a(0), a(1), ... given by x(n) = Phi(x(n-1)) (n>0), a(n) = P(x(n)) (n >= 0).

Costé prime expansion = Engel expansion where all terms must be primes (cf. A006784).

G. C. Greubel, Table of n, a(n) for n = 0..2000
A. Costé, Sur un système fibré lié à la suite des nombres premiers, Exper. Math., 11 (2002), 383-405.
Index entries for sequences related to Engel expansions

Cf. A079367, A079368. Also A079369-A079389.

Digits := 200: P := proc(x) local y; y := ceil(evalf(1/x)); if isprime(y) then y else nextprime(y); fi; end; F := proc(x) local y,i,t1; y := x; t1 := []; for i from 1 to 50 do p := P(y); t1 := [op(t1),p]; y := p*y-1; od; t1; end; F(Pi-3);

$MaxExtraPrecision = 40; P[x_] := Module[{y}, y = Ceiling[1/x]; If[PrimeQ[y], y, NextPrime[y]]]; F[x_] := Module[{y, i, t1}, y = x; t1 = {}; For[i = 1, i <= 70, i++, AppendTo[t1, p = P[y]]; y = p*y-1]; t1]; F[Pi-3] (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A079367 Records in A079366.

Original entry on oeis.org

11, 17, 23, 97, 107, 127, 409, 58889

Offset: 1

N. J. A. Sloane, Feb 15 2003

RECORDS transform of A079366.

N. J. A. Sloane, Transforms

Cf. A079366, A079368.

A079366 = Cases[Import["https://oeis.org/A079366/b079366.txt", "Table"], {, }][[All, 2]];
a = {}; l = 0;
For[i = 1, i <= Length[A079366], i++,
  If[A079366[[i]] > l, l = A079366[[i]]; AppendTo[a, l]]];
a (* Robert Price, Mar 14 2020 *)

A079385 Records in A079384.

Original entry on oeis.org

5, 53, 71, 379, 541, 1423, 2377, 3343, 9829, 13721

Offset: 1

N. J. A. Sloane, Feb 16 2003

RECORDS transform of A079384.

N. J. A. Sloane, Transforms

Cf. A079384, A079386, A079366, A079367, A079368.

A079384 = Cases[Import["https://oeis.org/A079384/b079384.txt", "Table"], {, }][[All, 2]];
a = {}; l = 0;
For[i = 1, i <= Length[A079384], i++,
  If[A079384[[i]] > l, l = A079384[[i]]; AppendTo[a, l]]];
a (* Robert Price, Mar 15 2020 *)

a(4)-a(10) from Robert Price, Mar 15 2020

A079386 Where records occur in A079384.

Original entry on oeis.org

1, 8, 21, 76, 163, 315, 471, 999, 1168, 1238

Offset: 1

N. J. A. Sloane, Feb 16 2003

RECORDS transform of A079384.

N. J. A. Sloane, Transforms

Cf. A079384, A079385, A079366, A079367, A079368.

A079384 = Cases[Import["https://oeis.org/A079384/b079384.txt", "Table"], {, }][[All, 2]];
a = {}; l = 0;
For[i = 1, i <= Length[A079384], i++,
  If[A079384[[i]] > l, l = A079384[[i]]; AppendTo[a, i]]];
a (* Robert Price, Mar 15 2020 *)

a(4)-a(10) from Robert Price, Mar 15 2020

A079369 Costé prime expansion of e - 2.

Original entry on oeis.org

2, 3, 5, 2, 11, 17, 11, 3, 37, 11, 11, 3, 5, 2, 11, 2, 11, 2, 53, 37, 7, 7, 73, 73, 79, 19, 17, 11, 5, 37, 7, 5, 7, 29, 277, 2903, 607, 211, 29, 11, 739, 9463, 8693, 3907, 307, 23, 223, 59, 37, 11, 2, 41, 23, 11, 3, 23, 7, 5, 5, 2, 11, 5, 7, 7, 5, 2, 5, 2, 5, 3, 67, 41, 223, 31, 107

Offset: 0

N. J. A. Sloane, Feb 16 2003

For x in (0,1], define P(x) = min{p: p prime, 1/x < p}, Phi(x) = P(x)x - 1. Costé prime expansion of x(0) is sequence a(0), a(1), ... given by x(n) = Phi(x(n-1)) (n>0), a(n) = P(x(n)) (n >= 0).

G. C. Greubel, Table of n, a(n) for n = 0..2000
A. Costé, Sur un système fibré lié à la suite des nombres premiers, Exper. Math., 11 (2002), 383-405.

Cf. A079370, A079371, A079366, A079367, A079368.

Digits := 500: P := proc(x) local y; y := ceil(evalf(1/x)); if isprime(y) then y else nextprime(y); fi; end; F := proc(x) local y,i,t1; y := x; t1 := []; for i from 1 to 100 do p := P(y); t1 := [op(t1),p]; y := p*y-1; od; t1; end; F(exp(1)-2);

$MaxExtraPrecision = 500; P[x_] := Module[{y}, y = Ceiling[1/x]; If[PrimeQ[y], y, NextPrime[y]]]; F[x_] := Module[{y, i, t1}, y = x; t1 = {}; For[i = 1, i <= 100, i++, AppendTo[t1, p = P[y]]; y = p*y - 1]; t1]; F[E - 2] (* G. C. Greubel, Jan 21 2019 *)

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 17 2003

A079372 Costé prime expansion of sqrt(2) - 1.

Original entry on oeis.org

3, 5, 5, 17, 11, 3, 29, 31, 29, 13, 7, 37, 7, 5, 3, 5, 5, 5, 11, 17, 7, 13, 13, 17, 11, 5, 3, 31, 31, 53, 41, 97, 47, 19, 17, 17, 41, 71, 29, 11, 211, 23, 79, 17, 5, 7, 23, 17, 5, 3, 11, 5, 2, 17, 7, 17, 5, 2, 23, 11, 3, 3, 5, 5, 3, 3, 5

Offset: 0

N. J. A. Sloane, Feb 16 2003

For x in (0,1], define P(x) = min{p: p prime, 1/x < p}, Phi(x) = P(x)x - 1. Costé prime expansion of x(0) is sequence a(0), a(1), ... given by x(n) = Phi(x(n-1)) (n>0), a(n) = P(x(n)) (n >= 0).

Nathaniel Johnston, Table of n, a(n) for n = 0..5000
A. Costé, Sur un système fibré lié à la suite des nombres premiers, Exper. Math., 11 (2002), 383-405.

Cf. A079373, A079374, A079366, A079367, A079368.

Digits := 200: P := proc(x) local y; y := ceil(evalf(1/x)); if isprime(y) then y else nextprime(y); fi; end; F := proc(x) local y,i,t1; y := x; t1 := []; for i from 1 to 50 do p := P(y); t1 := [op(t1),p]; y := p*y-1; od; t1; end; F(sqrt(2)-1);

$MaxExtraPrecision = 500; P[x_] := Module[{y}, y = Ceiling[1/x]; If[PrimeQ[y], y, NextPrime[y]]]; F[x_] := Module[{y, i, t1}, y = x; t1 = {}; For[i = 1, i <= 100, i++, AppendTo[t1, p = P[y]]; y = p*y - 1]; t1]; F[Sqrt[2] -1] (* G. C. Greubel, Jan 20 2019 *)

A079375 Costé prime expansion of Pi^2 - 9.

Original entry on oeis.org

2, 2, 3, 3, 5, 2, 19, 11, 29, 53, 149, 23, 7, 11, 5, 5, 3, 5, 5, 59, 11, 7, 7, 41, 19, 17, 23, 7, 5, 3, 7, 3, 11, 3, 3, 5, 2, 5, 3, 3, 53, 11, 5, 3, 41, 13, 29, 97, 13, 11, 2, 11, 5, 7, 7, 17, 5, 11, 3, 3, 7, 23, 53, 11, 5, 17, 5, 7, 3, 17

Offset: 0

N. J. A. Sloane, Feb 16 2003

For x in (0,1], define P(x) = min{p: p prime, 1/x < p}, Phi(x) = P(x)x - 1. Costé prime expansion of x(0) is sequence a(0), a(1), ... given by x(n) = Phi(x(n-1)) (n>0), a(n) = P(x(n)) (n >= 0).

Nathaniel Johnston, Table of n, a(n) for n = 0..5000
A. Costé, Sur un système fibré lié à la suite des nombres premiers, Exper. Math., 11 (2002), 383-405.

Cf. A079376, A079377, A079366, A079367, A079368.

Digits := 200: P := proc(x) local y; y := ceil(evalf(1/x)); if isprime(y) then y else nextprime(y); fi; end; F := proc(x) local y,i,t1; y := x; t1 := []; for i from 1 to 50 do p := P(y); t1 := [op(t1),p]; y := p*y-1; od; t1; end; F(Pi^2 - 9);

$MaxExtraPrecision = 500; P[x_] := Module[{y}, y = Ceiling[1/x]; If[PrimeQ[y], y, NextPrime[y]]]; F[x_] := Module[{y, i, t1}, y = x; t1 = {}; For[i = 1, i <= 100, i++, AppendTo[t1, p = P[y]]; y = p*y - 1]; t1]; F[Pi^2 -9] (* G. C. Greubel, Jan 20 2019 *)

A079378 Costé prime expansion of (sqrt(5)-1)/2.

Original entry on oeis.org

2, 5, 7, 5, 5, 2, 11, 5, 2, 13, 11, 5, 5, 5, 5, 2, 5, 3, 5, 11, 2, 79, 37, 11, 5, 3, 7, 17, 59, 23, 11, 2, 17, 7, 3, 7, 7, 383, 337, 67, 13, 19, 47, 23, 41, 17, 17, 13, 67, 71, 47, 17, 11, 19, 73, 53, 17, 13, 37, 19, 11, 5, 13, 29, 43, 47, 17, 5, 5, 11, 3, 3, 17, 5, 97, 29, 11, 3, 3, 3, 7

Offset: 0

N. J. A. Sloane, Feb 16 2003

For x in (0,1], define P(x) = min{p: p prime, 1/x < p}, Phi(x) = P(x)x - 1. Costé prime expansion of x(0) is sequence a(0), a(1), ... given by x(n) = Phi(x(n-1)) (n>0), a(n) = P(x(n)) (n >= 0).

G. C. Greubel, Table of n, a(n) for n = 0..2000
A. Costé, Sur un système fibré lié à la suite des nombres premiers, Exper. Math., 11 (2002), 383-405.

Cf. A079379, A079380, A079366, A079367, A079368.

Digits := 500: P := proc(x) local y; y := ceil(evalf(1/x)); if isprime(y) then y else nextprime(y); fi; end; F := proc(x) local y,i,t1; y := x; t1 := []; for i from 1 to 100 do p := P(y); t1 := [op(t1),p]; y := p*y-1; od; t1; end; F((sqrt(5)-1)/2);

$MaxExtraPrecision = 500; P[x_] := Module[{y}, y = Ceiling[1/x]; If[PrimeQ[y], y, NextPrime[y]]]; F[x_] := Module[{y, i, t1}, y = x; t1 = {}; For[i = 1, i <= 100, i++, AppendTo[t1, p = P[y]]; y = p*y - 1]; t1]; F[(Sqrt[5] -1)/2] (* G. C. Greubel, Jan 20 2019 *)

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 17 2003

A079381 Costé prime expansion of Euler's constant gamma (A001620).

Original entry on oeis.org

2, 7, 13, 19, 89, 23, 11, 131, 73, 43, 37, 7, 11, 3, 3, 3, 3, 3, 5, 2, 7, 61, 251, 41, 13, 11, 7, 23, 29, 5, 13, 11, 3, 67, 29, 7, 5, 5, 2, 17, 5, 23, 7, 11, 2, 31, 29, 5, 5, 5, 3, 3, 5, 11, 5, 7, 7, 29, 17, 5, 2, 41, 13, 13, 11, 199, 157, 101, 37, 7, 127, 29, 11, 3, 3, 5, 17, 5, 7, 5, 2, 5

Offset: 0

N. J. A. Sloane, Feb 16 2003

For x in (0,1], define P(x) = min{p: p prime, 1/x < p}, Phi(x) = P(x)x - 1. Costé prime expansion of x(0) is sequence a(0), a(1), ... given by x(n) = Phi(x(n-1)) (n>0), a(n) = P(x(n)) (n >= 0).

G. C. Greubel, Table of n, a(n) for n = 0..2000
A. Costé, Sur un système fibré lié à la suite des nombres premiers, Exper. Math., 11 (2002), 383-405.

Cf. A079382, A079383, A079366-A079368.

Digits := 500: P := proc(x) local y; y := ceil(evalf(1/x)); if isprime(y) then y else nextprime(y); fi; end; F := proc(x) local y,i,t1; y := x; t1 := []; for i from 1 to 100 do p := P(y); t1 := [op(t1),p]; y := p*y-1; od; t1; end; F(gamma);

$MaxExtraPrecision = 500; P[x_] := Module[{y}, y = Ceiling[1/x]; If[PrimeQ[y], y, NextPrime[y]]]; F[x_] := Module[{y, i, t1}, y = x; t1 = {}; For[i = 1, i <= 100, i++, AppendTo[t1, p = P[y]]; y = p*y - 1]; t1]; F[EulerGamma] (* G. C. Greubel, Jan 20 2019 *)

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 17 2003

A079384 Costé prime expansion of 2^(1/3) - 1.

Original entry on oeis.org

5, 5, 3, 3, 3, 3, 3, 53, 13, 29, 7, 5, 11, 3, 7, 5, 11, 5, 7, 5, 71, 67, 17, 11, 5, 5, 37, 11, 2, 11, 11, 3, 11, 5, 5, 11, 7, 11, 3, 3, 3, 3, 3, 3, 11, 7, 7, 7, 5, 23, 17, 7, 17, 13, 41, 53, 23, 7, 7, 5, 5, 5, 5, 2, 7, 3, 53, 11, 3, 7, 5, 2, 7, 17, 17, 379, 107, 41, 19, 11, 5, 5, 2, 5, 5, 3, 11, 2, 29, 11, 7, 7, 13, 29, 7, 53, 53, 17, 11, 3

Offset: 0

N. J. A. Sloane, Feb 16 2003

For x in (0,1], define P(x) = min{p: p prime, 1/x < p}, Phi(x) = P(x)x - 1. Costé prime expansion of x(0) is sequence a(0), a(1), ... given by x(n) = Phi(x(n-1)) (n>0), a(n) = P(x(n)) (n >= 0).

G. C. Greubel, Table of n, a(n) for n = 0..2000
A. Costé, Sur un système fibré lié à la suite des nombres premiers, Exper. Math., 11 (2002), 383-405.

Cf. A079385, A079386, A079366, A079367, A079368.

Digits := 500: P := proc(x) local y; y := ceil(evalf(1/x)); if isprime(y) then y else nextprime(y); fi; end; F := proc(x) local y,i,t1; y := x; t1 := []; for i from 1 to 100 do p := P(y); t1 := [op(t1),p]; y := p*y-1; od; t1; end; F(2^(1/3) - 1);

$MaxExtraPrecision = 500; P[x_]:= Module[{y}, y = Ceiling[1/x]; If[PrimeQ[y], y, NextPrime[y]]]; F[x_] := Module[{y, i, t1}, y = x; t1 = {}; For[i = 1, i <= 100, i++, AppendTo[t1, p = P[y]]; y = p*y - 1]; t1]; F[Surd[2, 3] - 1] (* G. C. Greubel, Jan 20 2019 *)

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 17 2003

cp's OEIS Frontend

A079366 Costé prime expansion of Pi - 3.

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A079367 Records in A079366.

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A079385 Records in A079384.

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A079386 Where records occur in A079384.

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A079369 Costé prime expansion of e - 2.

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A079372 Costé prime expansion of sqrt(2) - 1.

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A079375 Costé prime expansion of Pi^2 - 9.

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A079378 Costé prime expansion of (sqrt(5)-1)/2.

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Maple