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
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
Programs
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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
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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)
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