cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A317253 Indices m for which A317414(m) = 1.

Original entry on oeis.org

4, 8, 9, 11, 16, 17, 20, 22, 23, 27, 32, 33, 36, 40, 41, 43, 46, 47, 52, 54, 55, 59, 64, 65, 68, 72, 73, 75, 80, 81, 84, 86, 87, 91, 94, 95, 100, 104, 105, 107, 110, 111, 116, 118, 119, 123, 128, 129, 132, 136, 137, 139, 144, 145, 148, 150, 151, 155, 160, 161
Offset: 1

Views

Author

A.H.M. Smeets, Aug 13 2018

Keywords

Comments

This is a supporting sequence for A317627.

Crossrefs

Cf. A317414, A317627.

A058304 Continued fraction for Liouville's number (A012245).

Original entry on oeis.org

0, 9, 11, 99, 1, 10, 9, 999999999999, 1, 8, 10, 1, 99, 11, 9, 999999999999999999999999999999999999999999999999999999999999999999999999, 1, 8, 11, 99, 1, 10, 8, 1, 999999999999, 9, 10, 1, 99, 11, 9
Offset: 0

Views

Author

Robert G. Wilson v, Dec 08 2000

Keywords

Comments

From A.H.M. Smeets, Jun 06 2018: (Start)
Except for the first term, the only values that occur in this sequence are 1,8,9,10,11,and values 10^((m-1)*m!)-1 for m > 1. The probability of occurrence P(a(n) = k) are given by:
P(a(n) = 1) = 1/4,
P(a(n) = 8) = 1/8,
P(a(n) = 9) = 1/8,
P(a(n) = 10) = 1/8,
P(a(n) = 11) = 1/8 and
P(a(n) = 10^((m-1)*m!)-1) = 2^-(m+1) for m > 1. (End)

Examples

			0.1100010000000000000000010... = 0 + 1/(9 + 1/(11 + 1/(99 + 1/(1 + ...)))). - _Harry J. Smith_, May 15 2009

References

  • Harold M. Stark, "An Introduction to Number Theory," The MIT Press, Cambridge, MA and London, England, Eighth Printing, 1994, pages 172 - 177.

Links

Crossrefs

Cf. A012245.
Cf. A317413 (in base 2), A317414 (in base 3) A317661 (in base 4 and general).

Programs

  • Maple
    with(numtheory): cfrac(add(1/10^factorial(n),n=1..7),62,'quotients'); # Muniru A Asiru, Aug 08 2018
  • Mathematica
    ContinuedFraction[ Sum[ 1 /10^(n!), {n, 1, 7} ], 40 ]
  • PARI
    { allocatemem(932245000); default(realprecision, 200000); x=contfrac(suminf(n=1, 1.0/10^n!)); for (n=1, 255, write("b058304.txt", n, " ", x[n])); } \\ Harry J. Smith, May 15 2009
  • Python
    n,f,i,p,q,base = 1,1,0,0,1,10
while i < 1000:
    i,p,q = i+1,p*base,q*base
    if i == f:
        p,n = p+1,n+1
        f = f*n
n,a,j = 0,0,0
while p%q > 0:
    a,f,p,q = a+1,p//q,q,p%q
    print(a-1,f)
# A.H.M. Smeets, Aug 03 2018

Formula

From A.H.M. Smeets, Jun 26 2018: (Start)
a(n) = 1 iff n in A317331,
a(n) = 8 iff n in A317332,
a(n) = 9 iff n in A317333,
a(n) = 10 iff n = 8*m - 6 + 3*(m mod 2) for m > 0,
a(n) = 11 iff n = 8*m - 3 - 3*(m mod 2) for m > 0,
a(n) = 10^((m-1)*m!)-1 iff n in {2^m*(1+k*4) - 1 | k >= 0} union {2^m*(3+k*4) | k >= 0} for m > 1. (End)

Extensions

Offset changed to 0 on the advice of A.H.M. Smeets by Muniru A Asiru, Aug 11 2018

A317413 Continued fraction for binary expansion of Liouville's number interpreted in base 2 (A092874).

Original entry on oeis.org

0, 1, 3, 3, 1, 2, 1, 4095, 3, 1, 3, 3, 1, 4722366482869645213695, 4, 3, 1, 3, 4095, 1, 2, 1, 3, 3, 1
Offset: 0

Views

Author

A.H.M. Smeets, Jul 27 2018

Keywords

Comments

The continued fraction of the number obtained by reading A012245 as binary fraction.
Except for the first term, the only values that occur in this sequence are 1, 2, 3, 4 and values 2^((m-1)*m!) - 1 for m > 2. The probability of occurrence P(a(n) = k) are given by:
P(a(n) = 1) = 1/3,
P(a(n) = 2) = 1/12,
P(a(n) = 3) = 1/3,
P(a(n) = 4) = 1/12 and
P(a(n) = 2^((m-1)*m!)-1) = 1/(3*2^(m-1)) for m > 2.
The next term is roughly 3.12174855*10^144 (see b-file for precise value).

Examples

			0.76562505... = 0+1/(1+1/(3+1/(3+1/(1+1/(2+...))))). - _R. J. Mathar_, Jun 19 2021

Links

Crossrefs

Cf. A012245, A014710, A317538, A317539.
Cf. A058304 (in base 10), A317414 (in base 3).

Programs

  • Maple
    with(numtheory): cfrac(add(1/2^factorial(n),n=1..7),24,'quotients'); # Muniru A Asiru, Aug 11 2018
  • Mathematica
    ContinuedFraction[ FromDigits[ RealDigits[ Sum[1/10^n!, {n, 8}], 10, 10000], 2], 60] (* Robert G. Wilson v, Aug 09 2018 *)
  • Python
    n,f,i,p,q,base = 1,1,0,0,1,2
while i < 100000:
    i,p,q = i+1,p*base,q*base
    if i == f:
        p,n = p+1,n+1
        f = f*n
n,a,j = 0,0,0
while p%q > 0:
    a,f,p,q = a+1,p//q,q,p%q
    print(a-1,f)

Formula

a(n) = 1 if and only if n in A317538.
a(n) = 2 if and only if n in {24*m - 19 | m > 0} union {24*m - 4 | m > 0}.
a(n) = 3 if and only if n in A317539.
a(n) = 4 if and only if n in {12*m + A014710(m-1) - 2*(A014710(m-1) mod 2) | m > 0}
a(n) = 2^((m-1)*m!)-1 if and only if n in {3*2^(m-2)*(1+k*4) + 1 | k >= 0} union {3*2^(m-2)*(3+k*4) | k >= 0} for m > 2.

A317661 Continued fraction for quaternary expansion of Liouville's number interpreted in base 4 (A012245).

Original entry on oeis.org

0, 3, 5, 15, 1, 4, 3, 16777215, 1, 2, 4, 1, 15, 5, 3, 22300745198530623141535718272648361505980415, 1, 2, 5, 15, 1, 4, 2, 1, 16777215, 3, 4, 1, 15, 5, 3
Offset: 0

Views

Author

A.H.M. Smeets, Aug 03 2018

Keywords

Links

Crossrefs

Cf. A012245, A317331, A317332, A317333.
Cf. A058304 (in base 10), A317413 (in base 2), A317414 (in base 3).

Programs

  • Maple
    with(numtheory): cfrac(add(1/4^factorial(n),n=1..7),30,'quotients'); # Muniru A Asiru, Aug 12 2018
  • Python
    n, f, i, p, q, base = 1, 1, 0, 0, 1, 4
while i < 100000:
    i, p, q = i+1, p*base, q*base
    if i == f:
        p, n = p+1, n+1
        f = f*n
n, a, j = 0, 0, 0
while p%q > 0:
    a, f, p, q = a+1, p//q, q, p%q
    print(a-1, f)

Formula

In general for any Liouville's number base > 2:
a(n) = 1 if (and only if, for base > 3) n in A317331,
a(n) = base-2 if (and only if, for base > 3) n in A317332,
a(n) = base-1 if and only if n in A317333,
a(n) = base if and only if n in {8*m - 6 + 3*(m mod 2) | m > 0},
a(n) = base+1 if and only if n in {8*m - 3 - 3*(m mod 2) | m > 0},
a(n) = base^((m-1)*m!)-1 iff n in {2^m*(1+k*4) - 1 | k >= 0} union {2^m*(3+k*4) | k >= 0} for m > 1.

A317627 a(n) = A317253 - floor(8*n/3).

Original entry on oeis.org

2, 3, 1, 1, 3, 1, 2, 1, -1, 1, 3, 1, 2, 3, 1, 1, 1, -1, 2, 1, -1, 1, 3, 1, 2, 3, 1, 1, 3, 1, 2, 1, -1, 1, 1, -1, 2, 3, 1, 1, 1, -1, 2, 1, -1, 1, 3, 1, 2, 3, 1, 1, 3, 1, 2, 1, -1, 1, 3, 1, 2, 3, 1, 1, 1, -1, 2, 1, -1, 1, 1, -1, 2, 3, 1, 1, 3, 1, 2
Offset: 1

Views

Author

A.H.M. Smeets, Aug 02 2018

Keywords

Crossrefs

Cf. A000034, A014577, A034947,A089607, A317414.

Programs

  • Python
    n,f,i,p,q,base = 1,1,0,0,1,3
while i < 100000:
    i,p,q = i+1,p*base,q*base
    if i == f:
        p,n = p+1,n+1
        f = f*n
n,a,j = 0,0,0
while p%q > 0:
    a,f,p,q = a+1,p//q,q,p%q
    if f == 1:
        n = n+1
        print(n,a-1-(8*n//3))

Formula

a(3*n+1) = 2 - (n mod 2) for n >= 0, a(6*n+2) = 3 - 2*(n mod 2) and a(6*n+5) = a(3*n+2) for n >= 0, a(6*n+3) = 1 - 2*(n mod 2) and a(6*n+6) = a(3*n+3) for n >= 0.
a(3*n+1) = A000034(n+1) for n >= 0.
a(3*n+2) = A089607(n) for n > 1.
a(3*n+2) = 2*A014577(n-1)+1 for n > 0.
a(3*n+3) = A034947(n) = 2*A014577(n-1)-1 for n > 0.
Showing 1-5 of 5 results.

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