A317538 -id:A317538 - cp's OEIS frontend

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

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

A.H.M. Smeets, Jul 27 2018

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).

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

A.H.M. Smeets, Table of n, a(n) for n = 0..48

Cf. A012245, A014710, A317538, A317539.

Cf. A058304 (in base 10), A317414 (in base 3).

with(numtheory): cfrac(add(1/2^factorial(n),n=1..7),24,'quotients'); # Muniru A Asiru, Aug 11 2018

ContinuedFraction[ FromDigits[ RealDigits[ Sum[1/10^n!, {n, 8}], 10, 10000], 2], 60] (* Robert G. Wilson v, Aug 09 2018 *)

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)

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.

A317539 Indices m for which A317413(m) = 3.

Original entry on oeis.org

2, 3, 8, 10, 11, 15, 17, 22, 23, 27, 32, 34, 38, 39, 41, 46, 47, 51, 56, 58, 59, 63, 65, 70, 74, 75, 80, 82, 86, 87, 89, 94, 95, 99, 104, 106, 107, 111, 113, 118, 119, 123, 128, 130, 134, 135, 137, 142, 146, 147, 152, 154, 155, 159, 161, 166, 170, 171, 176, 178, 182, 183, 185, 190, 191, 195, 200, 202

Offset: 1

A.H.M. Smeets, Jul 30 2018

nonn

A.H.M. Smeets, Table of n, a(n) for n = 1..20000

Cf. A317413, A317538, A317540.

a(n) = 3*(n-1) + A317540(n).

A359838 Continued fraction for binary expansion of A359456 interpreted in base 2.

Original entry on oeis.org

0, 1, 3, 3, 1, 2, 1, 262143, 3, 1, 3, 3, 1, 1532495540865888858358347027150309183618739122183602175, 4, 3, 1, 3, 262143, 1, 2, 1, 3, 3, 1

Offset: 0

A.H.M. Smeets, Jan 14 2023

The continued fraction of the number obtained by reading A359456 as a binary fraction.
Except for the first term, the only values that occur in this sequence are 1, 2, 3, 4 and values 2^A359458(m) - 1 for m > 2. The probabilities 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^A359458(m)-1) = 1/(3*2^m) for m > 1.

Cf. A014710, A317538, A317539, A359456, A359458.

Cf. A359457 (in base 10).

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 - 3*A014710(m-1) + 5 | m > 0}
a(n) = 2^A359458(m)-1 if and only if n in {3*2^(m-1)*(1+k*4) + 1 | k >= 0} union {3*2^(m-1)*(3+k*4) | k >= 0} for m > 1.

cp's OEIS Frontend

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

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A317539 Indices m for which A317413(m) = 3.

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A359838 Continued fraction for binary expansion of A359456 interpreted in base 2.

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