A058304 -id:A058304 - cp's OEIS frontend

A317331 Indices m for which A058304(m) = 1.

4, 8, 11, 16, 20, 23, 27, 32, 36, 40, 43, 47, 52, 55, 59, 64, 68, 72, 75, 80, 84, 87, 91, 95, 100, 104, 107, 111, 116, 119, 123, 128, 132, 136, 139, 144, 148, 151, 155, 160, 164, 168, 171, 175, 180, 183, 187, 191, 196, 200, 203, 208, 212, 215, 219, 223, 228, 232, 235, 239, 244, 247, 251

Offset: 1

A.H.M. Smeets, Jul 26 2018

nonn

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

Cf. A014577, A058304, A317332, A317333, A317335, A317336.

n, f, i, p, q = 1, 1, 0, 0, 1
while i < 100000:
    i, p, q = i+1, p*10, q*10
    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)

a(n) = 4*n + A014577(n-1)-1. - A.H.M. Smeets, Jul 29 2018

A317332 Indices m for which A058304(m) = 8.

Original entry on oeis.org

9, 17, 22, 33, 41, 46, 54, 65, 73, 81, 86, 94, 105, 110, 118, 129, 137, 145, 150, 161, 169, 174, 182, 190, 201, 209, 214, 222, 233, 238, 246, 257, 265, 273, 278, 289, 297, 302, 310, 321, 329, 337, 342, 350, 361, 366, 374, 382, 393, 401, 406, 417, 425, 430, 438, 446, 457, 465, 470, 478, 489, 494, 502

Offset: 1

A.H.M. Smeets, Jul 26 2018

nonn

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

Cf. A058304, A317331, A317333, A317335, A317336.

n, f, i, p, q = 1, 1, 0, 0, 1
while i < 1000000:
    i, p, q = i+1, p*10, q*10
    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 == 8:
        n = n+1
        print(n, a-1)

a(n) = 8*n + A317335(n).

A317333 Indices m for which A058304(m) = 9.

Original entry on oeis.org

1, 6, 14, 25, 30, 38, 49, 57, 62, 70, 78, 89, 97, 102, 113, 121, 126, 134, 142, 153, 158, 166, 177, 185, 193, 198, 206, 217, 225, 230, 241, 249, 254, 262, 270, 281, 286, 294, 305, 313, 318, 326, 334, 345, 353, 358, 369, 377, 385, 390, 398, 409, 414, 422, 433, 441, 449, 454, 462, 473, 481, 486, 497, 505

Offset: 1

A.H.M. Smeets, Jul 26 2018

nonn

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

Cf. A058304, A317331, A317332, A317335, A317336.

n, f, i, p, q = 1, 1, 0, 0, 1
while i < 1000000:
    i, p, q = i+1, p*10, q*10
    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 == 9:
        n = n+1
        print(n, a-1)

a(n) = 8*n + A317336(n).

A012245 Characteristic function of factorial numbers; also decimal expansion of Liouville's number or Liouville's constant.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

N. J. A. Sloane

Read as decimal fraction 1100010... in any base > 1 (arbitrary decimal point) Liouville's numbers are transcendental; read as a continued fraction it is also transcendental [G. H. Hardy and E. M. Wright, Th. 192].

			a(25) = a(26) = ... = a(119) = 0 because 4! = 24 and 5! = 120.
0.110001000000000000000001000000000000000000000000000000000000000000000....

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 89.
John H. Conway and Richard K. Guy, The Book of Numbers, pp. 239-241 (1996).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 162.
T. W. Koerner, Fourier Analysis, Camb. Univ. Press 1988, p. 177.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 58.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 26.

Harry J. Smith, Table of n, a(n) for n = 1..20000
J. Liouville, Communication, C. R. Acad. Sci. Paris 18, 883-885 and 993-995, 1844. [Pages 993-995 do not seem right]
J. Liouville, Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques, Journal de Mathématiques Pures et Appliquées 16, pp. 133-142, 1851.
Diego Marques and Carlos Gustavo Moreira, On variations of the Liouville constant which are also Liouville numbers, Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 3 (2016), 39-40.
Michael Penn, One of the first transcendental numbers -- Liouville's Constant, YouTube video, 2022.
Burkard Polster, Liouville's number, the easiest transcendental and its clones, Mathologer video (2017).
Eric Weisstein's World of Mathematics, Liouville's Constant.
G. Xiao, Contfrac.
Index entries for characteristic functions.
Index entries for continued fractions for constants.
Index entries for transcendental numbers.

Cf. A000142, A058304 (continued fraction).

With[{nn=5},ReplacePart[Table[0,{nn!}],Table[{n!},{n,nn}]->1]] (* Harvey P. Dale, Jul 22 2012 *)
RealDigits[ Sum[1/10^n!, {n, 5}], 10, 105][[1]] (* Robert G. Wilson v, Aug 03 2018 *)
CoefficientList[1/x Sum[x^k!, {k, 1, 5}], x] (* Jean-François Alcover, Nov 02 2018 *)

default(realprecision, 20080); x=10*suminf(n=1, 1.0/10^n!) + 1/10^20040; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b012245.txt", n, " ", d)); \\ Harry J. Smith, May 15 2009

from itertools import count
def A012245(n):
    c = 1
    for i in count(1):
        if (c:=c*i) >= n:
            return int(c==n) # Chai Wah Wu, Jan 11 2023

G.f.: Sum_{i>=1} x^Product_{j=1..i} j. - Jon Perry, Mar 31 2004

a(A000142(n)) = 1; a(A063992(n)) = 0. - Reinhard Zumkeller, Oct 11 2008

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

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.

A317414 Continued fraction for ternary expansion of Liouville's number interpreted in base 3 (A012245).

Original entry on oeis.org

0, 2, 4, 8, 1, 3, 2, 531440, 1, 1, 3, 1, 8, 4, 2, 22528399544939174411840147874772640, 1, 1, 4, 8, 1, 3, 1, 1, 531440, 2, 3, 1, 8, 4, 2

Offset: 0

A.H.M. Smeets, Jul 27 2018

The continued fraction of the number obtained by reading A012245 as a ternary fraction.
Except for the first term, the only values that occur in this sequence are 1,2,3,4, and values 3^((m-1)*m!)-1 for m > 1. The probability of occurrence P(a(n) = k) are given by:
P(a(n) = 1) = 3/8,
P(a(n) = 2) = 1/8,
P(a(n) = 3) = 1/8,
P(a(n) = 4) = 1/8 and
P(a(n) = 3^((m-1)*m!)-1) = 2^-(m+1) for m > 1.
More generally it seems that for any base > 2, P(a(n) <= base+1) = 3/4, P(a(n) > base+1) = 1/4, and P(a(n) = base^((m-1)*m!)-1) = 2^-(m+1) for m > 1.

A.H.M. Smeets, Table of n, a(n) for n = 0..62

Cf. A012245, A089013, A317627.

Cf. A058304 (in base 10), A317413 (in base 2), A317661 (in base 4).

with(numtheory): cfrac(add(1/3^factorial(n),n=1..7),30,'quotients'); # Muniru A Asiru, Aug 11 2018

ContinuedFraction[ FromDigits[ RealDigits[ Sum[1/10^n!, {n, 8}], 10, 10000], 3], 60] (* Robert G. Wilson v, Aug 09 2018 *)

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
    print(a-1,f)

a(n) = 1 if and only if n in {floor(8*n/3) + A317627(n) | n > 0}.
a(n) = 2 if and only if n in {8*n - 10 + 3*A089013(n-1) | n > 0}.
a(n) = 3 if and only if n in {16*n - 11 | n > 0} union {16*n - 6 | n > 0}.
a(n) = 4 if and only if n in {16*n - 14 | n > 0} union {16*n - 3 | n > 0}.
a(n) = 3^((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.

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

A.H.M. Smeets, Aug 03 2018

A.H.M. Smeets, Table of n, a(n) for n = 0..62

Cf. A012245, A317331, A317332, A317333.

Cf. A058304 (in base 10), A317413 (in base 2), A317414 (in base 3).

with(numtheory): cfrac(add(1/4^factorial(n),n=1..7),30,'quotients'); # Muniru A Asiru, Aug 12 2018

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)

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.

A359457 Continued fraction for constant A359456.

Original entry on oeis.org

0, 9, 11, 99, 1, 10, 9, 999999999999999999, 1, 8, 10, 1, 99, 11, 9

Offset: 0

A.H.M. Smeets, Jan 02 2023

It seems that all terms smaller than 100 are identical to the continued fraction terms of Liouville's constant as in A058304.
Except for the first term, the only values that occur in this sequence are 1,8,9,10,11,and values 10^A359458(m-1)-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^A359458(m-1)-1) = 2^-(m+1) for m > 1.

Cf. A058304, A317331, A317332, A317333, A359456, A359458.

a(n) = 1 if and only if n in A317331,
a(n) = 8 if and only if n in A317332,
a(n) = 9 if and only if n in A317333,
a(n) = 10 if and only if n = 8*m - 6 + 3*(m mod 2) for m > 0,
a(n) = 11 if and only if n = 8*m - 3 - 3*(m mod 2) for m > 0,
a(n) = 10^A359458(m-1)-1 if and only if n in {2^m*(1+k*4) - 1 | k >= 0} union {2^m*(3+k*4) | k >= 0} for m > 1.

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A317331 Indices m for which A058304(m) = 1.

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A317332 Indices m for which A058304(m) = 8.

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A317333 Indices m for which A058304(m) = 9.

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A012245 Characteristic function of factorial numbers; also decimal expansion of Liouville's number or Liouville's constant.

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

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A317414 Continued fraction for ternary expansion of Liouville's number interpreted in base 3 (A012245).

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

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A359457 Continued fraction for constant A359456.

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