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

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

Author

Eric W. Weisstein

Keywords

• nonn
• cons

Comments

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

Examples

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

Links

• 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.

Crossrefs

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

Cf. A036680 (Egyptian fractions).

Programs

Maple

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

Mathematica

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 *)

PARI

{ 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

PARI

{ 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

PARI

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

Formula

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)