A092874 -id:A092874 - 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.

A051006 Prime constant: decimal value of (A010051 interpreted as a binary number).

Original entry on oeis.org

4, 1, 4, 6, 8, 2, 5, 0, 9, 8, 5, 1, 1, 1, 1, 6, 6, 0, 2, 4, 8, 1, 0, 9, 6, 2, 2, 1, 5, 4, 3, 0, 7, 7, 0, 8, 3, 6, 5, 7, 7, 4, 2, 3, 8, 1, 3, 7, 9, 1, 6, 9, 7, 7, 8, 6, 8, 2, 4, 5, 4, 1, 4, 4, 8, 8, 6, 4, 0, 9, 6, 0, 6, 1, 9, 3, 5, 7, 3, 3, 4, 1, 9, 6, 2, 9, 0, 0, 4, 8, 4, 2, 8, 4, 7, 5, 7, 7, 7, 9, 3, 9, 6, 1, 6

Offset: 0

Eric W. Weisstein

From Ferenc Adorjan (fadorjan(AT)freemail.hu): (Start)
Decimal expansion of the representation of the sequence of primes by a single real in (0,1).
Any monotonic integer sequence can be represented by a real number in (0, 1) in such a way that in the binary representation of the real, the n-th digit of the fractional part is 1 if and only if n is in the sequence.
Examples of the inverse mapping are A092855 and A092857. (End)
Is the prime constant an EL number? See Chow's 1999 article. - Lorenzo Sauras Altuzarra, Oct 05 2020
The asymptotic density of numbers with a prime number of trailing 0's in their binary representation (A370596), or a prime number of trailing 1's. - Amiram Eldar, Feb 23 2024

			0.414682509851111660... (base 10) = .01101010001010001010001... (base 2).

Harry J. Smith, Table of n, a(n) for n = 0..20000
Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function.
Timothy Y. Chow, What is a Closed-Form Number?, The American Mathematical Monthly, Vol. 106, No. 5. (May, 1999), pp. 440-448.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
Michael Penn, What is the prime constant and is it irrational?, YouTube video, 2022.
Simon Plouffe, Primes coded in binary to 1000 digits.
Eric Weisstein's World of Mathematics, Prime Constant.

Cf. A000720, A010051, A034785, A051007, A132800, A092857, A092858, A092859, A092860, A092861, A092862, A092863, A092874, A119523, A370596.

Cf. A036680 (Egyptian fractions).

a := n -> ListTools:-Reverse(convert(floor(evalf[1000](sum(1/2^ithprime(k), k = 1 .. infinity)*10^(n+1))), base, 10))[n+1]: - Lorenzo Sauras Altuzarra, Oct 05 2020

RealDigits[ FromDigits[ {{Table[ If[ PrimeQ[n], 1, 0], {n, 370}]}, 0}, 2], 10, 111][[1]] (* Robert G. Wilson v, Jan 15 2005 *)
RealDigits[Sum[1/2^Prime[k], {k, 1000}], 10, 100][[1]] (* Alexander Adamchuk, Aug 22 2006 *)

{ mt(v)= /*Returns the binary mapping of v monotonic sequence as a real in (0,1)*/ local(a=0.0,p=1,l);l=matsize(v)[2]; for(i=1,l,a+=2^(-v[i])); return(a)} \\ Ferenc Adorjan

{ default(realprecision, 20080); x=0; m=67000; for (n=1, m, if (isprime(n), a=1, a=0); x=2*x+a; ); x=10*x/2^m; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b051006.txt", n, " ", d)); } \\ Harry J. Smith, Jun 15 2009

suminf(n=1,.5^prime(n)) \\ Then: digits(%\.1^default(realprecision)) to get seq. of digits. N.B.: Functions sumpos() and sumnum() yield much less accurate results. - M. F. Hasler, Jul 04 2017

Prime constant C = Sum_{k>=1} 1/2^prime(k), where prime(k) is the k-th prime. - Alexander Adamchuk, Aug 22 2006
From Amiram Eldar, Aug 11 2020: (Start)
Equals Sum_{k>=1} A010051(k)/2^k.
Equals Sum_{k>=1} 1/A034785(k).
Equals (1/2) * A119523.
Equals Sum_{k>=1} pi(k)/2^(k+1), where pi(k) = A000720(k). (End)

A050923 a(n) = 2^(n!).

Original entry on oeis.org

2, 2, 4, 64, 16777216, 1329227995784915872903807060280344576

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 30 1999

For n > 0, every n-fold repetition of a(n) is a "powerful" arithmetic progression with difference 0; e.g., for n = 4 we get a(4) = 16777216 and in the generated repeating sequence of length 4 the k-th term is a k-th power (1 <= k <= n): 16777216 = 16777216^1, 16777216 = 4096^2, 16777216 = 256^3, 16777216 = 64^4. - Martin Renner, Aug 16 2017
From Jianing Song, Jul 20 2021: (Start)
Let F_q be the finite field with q elements, then in F_a(n), every polynomial of degree at most n splits into linear factors.
Union_{n>=0} F_a(n) is the algebraic clousre of F_2, which is the unique algebraically closed field with characteristic 2 and transcendence degree 0 (note that an algebraically closed field is uniquely determined by its characteristic and transcendence degree). Union_{n>=0} F_(2^lcm(1,2,...,n)) = Union_{n>=0} F_A178981(n) gives the same field.
Obviously, here 2 can be replaced by any prime p provided that {a(n)} is defined as a(n) = p^(n!). (End)
For n >= 1, the number of digits of a(n) is A317873(n). - Martin Renner, Mar 24 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..6
Index to divisibility sequences

Cf. A000079, A000142, A092874, A100731, A178981, A317873.

[2^Factorial(n): n in [0..8]]; // Vincenzo Librandi, Dec 16 2012

a=2;lst={};Do[a=a^n;AppendTo[lst,a],{n,1,7}];lst (* Vladimir Joseph Stephan Orlovsky, May 26 2009 *)
Table[2^n!,{n,0,9}] (* Vincenzo Librandi, Dec 16 2012 *)

makelist(2^(n!),n,0,5); /* Martin Ettl, Dec 27 2012 */

a(n)=2^n! \\ Charles R Greathouse IV, Aug 16 2017

a(n) = a(n-1)^n, a(0)=2.
a(n) = A000079(A000142(n)).
Sum_{n>=1} 1/a(n) = A092874. - Amiram Eldar, Oct 27 2020

A092858 "Sum" of the sequences of primes and the triangular numbers (A000217).

Original entry on oeis.org

5, 6, 7, 10, 11, 13, 15, 17, 19, 21, 23, 28, 29, 31, 36, 37, 41, 43, 45, 47, 53, 55, 59, 61, 66, 67, 71, 73, 78, 79, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 120, 127, 131, 136, 137, 139, 149, 151, 153, 157, 163, 167, 171, 173, 179, 181, 190, 191, 193, 197, 199

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

If two monotonic sequences are mapped into the real codomain of (0,1) as it is defined in A051006, then the fractional part of the sum of the two reals can be mapped back into a sequence as defined in A092855, yielding the "sum" of the two sequences.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092859, A092860, A092861, A092862, A092863, A092874.

{ssum(a,b)= /*Returns the "sum" monotonic sequences a and b */ return(mtinv(mt(a)+mt(b))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092859 "Difference" of the sequences of triangular numbers (A000217) and the primes (cf. A092858).

Original entry on oeis.org

3, 4, 5, 7, 12, 13, 16, 18, 19, 22, 23, 30, 31, 38, 39, 40, 42, 43, 46, 48, 49, 50, 51, 52, 53, 56, 57, 58, 60, 61, 68, 69, 70, 72, 73, 80, 81, 82, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 99, 100, 102, 103, 106, 108, 110, 111, 112, 113, 121, 122, 123, 124, 125, 126

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Here the complement of the sequence of primes (1 and the composites) is "added" to the sequence of triangulars, according to the definition outlined in A092858.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092860, A092861, A092862, A092863, A092874.

{sdif(a,b)= /*Returns the "difference" of monotonic sequences a and b */ return(mtinv(mt(a)+mt(compl(b)))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ } {compl(v)=/* Returns the complement of v monotonic positive sequence */ local(n,p=0,vv=[]);n=matsize(v)[2];for(i=1,n, for(j=p+1,v[i]-1,vv=concat (vv,j));p=v[i]);return(vv)}

A092860 "3 times the prime sequence".

Original entry on oeis.org

3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 28, 29, 30, 31, 36, 37, 40, 41, 42, 43, 46, 47, 52, 53, 58, 59, 60, 61, 66, 67, 70, 71, 72, 73, 78, 79, 82, 83, 88, 89, 96, 97, 100, 101, 102, 103, 106, 107, 108, 109, 112, 113, 126, 127, 130, 131, 136, 137, 138, 139

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

By iterating the addition to itself a monotonic sequence, according to the definition given in A092858, we can multiply the monotonic sequences by natural numbers.

Note, that it is easy to see that for an i natural and a v monotonic sequence, i(x)compl(v)=compl(i(x)v); where the "(x)" mark stands for the "integer multiplication of a sequence" and the function "compl" produces the complement of a positive monotonic sequence.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092859, A092861, A092862, A092863, A092874.

{imulv(i,v)= /*Returns "i (x) v" monotonic sequence */ return(mtinv(i*mt(v))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092861 "Product" of the sequence of primes and the "evil" numbers (A001969).

Original entry on oeis.org

4, 7, 9, 12, 14, 15, 18, 19, 21, 25, 26, 33, 35, 36, 37, 40, 41, 42, 44, 47, 48, 50, 54, 55, 58, 59, 60, 64, 65, 66, 69, 72, 77, 78, 79, 80, 84, 86, 87, 88, 89, 90, 91, 97, 99, 100, 105, 106, 107, 108, 110, 111, 112, 114, 115, 116, 117, 118, 120, 122, 123, 125, 127, 128

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

If two monotonic sequences are mapped into the real section of (0,1), as it is defined in A051006 and the product of the two reals mapped back into the set of monotonic sequences as defined in A092855, then we have the "product" of the two sequences.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092859, A092860, A092862, A092863, A092874.

{prod(a,b)= /*Returns the "product" of monotonic sequences a and b */ return(mtinv(mt(a)*mt(b))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092862 "Square" of the prime sequence.

Original entry on oeis.org

3, 5, 6, 14, 16, 17, 19, 21, 22, 25, 27, 31, 32, 34, 36, 37, 41, 42, 44, 45, 48, 49, 52, 54, 57, 59, 60, 62, 64, 65, 69, 74, 75, 78, 81, 88, 90, 91, 92, 94, 97, 98, 100, 103, 104, 108, 109, 114, 118, 119, 121, 123, 124, 125, 127, 128, 129, 130, 131, 133, 135, 136, 137

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

By following the definition outlined in A092861, one can multiply a monotonic sequence by itself, thus squaring it.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092859, A092860, A092861, A092863, A092874.

{pow(a,n)= /*Returns the "n-th power" of monotonic sequence a */ return(mtinv(mt(a)^n)) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092863 Prime sequence to the power Pi.

Original entry on oeis.org

4, 7, 10, 16, 18, 20, 22, 23, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 40, 42, 46, 51, 57, 60, 65, 66, 67, 68, 69, 70, 72, 73, 74, 77, 78, 80, 81, 82, 84, 85, 89, 91, 92, 93, 94, 95, 99, 101, 103, 107, 108, 110, 111, 112, 115, 117, 122, 123, 124, 125, 127, 128, 129, 130

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

F. Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092859, A092860, A092861, A092862, A092874.

{prow(a,r)= /*Returns the "r-th power" of monotonic sequence a */ return(mtinv(mt(a)^r)) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A076187 Decimal expansion of Sum_{k>=0} 1/2^(k!).

Original entry on oeis.org

1, 2, 6, 5, 6, 2, 5, 0, 5, 9, 6, 0, 4, 6, 4, 4, 7, 7, 5, 3, 9, 0, 6, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 5, 2, 3, 1, 6, 3, 8, 4, 5, 2, 6, 2, 6, 4, 0, 0, 5, 0, 9, 9, 9, 9, 1, 3, 8, 3, 8, 2, 2, 2, 3, 7, 2, 3, 3, 8, 0, 3, 9, 4, 5, 9, 5, 6, 3, 3, 4, 1, 3, 6, 0, 1, 3, 7, 6, 5, 6, 0, 1, 0, 9, 2, 0, 1, 8, 1, 8

Offset: 1

Benoit Cloitre, Nov 02 2002

			1.2656250596046447753906250000000000007...

Cf. A050923, A092874.

digits = 105; NSum[ 1/2^k!, {k, 0, 12} , WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 04 2013 *)
RealDigits[Total[1/2^Range[0,10]!],10,120][[1]] (* Harvey P. Dale, Oct 16 2024 *)

suminf(k=0, 1/2^(k!)) \\ Michel Marcus, Feb 19 2021

Equals Sum_{k>=0} 1/A050923(k).

Equals A092874 plus 0.5. [R. J. Mathar, Sep 08 2008]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A051006 Prime constant: decimal value of (A010051 interpreted as a binary number).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A050923 a(n) = 2^(n!).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A092858 "Sum" of the sequences of primes and the triangular numbers (A000217).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A092859 "Difference" of the sequences of triangular numbers (A000217) and the primes (cf. A092858).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A092860 "3 times the prime sequence".

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A092861 "Product" of the sequence of primes and the "evil" numbers (A001969).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A092862 "Square" of the prime sequence.

Views