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%I #76 Feb 23 2024 18:28:34
%S 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,
%T 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,
%U 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
%N Prime constant: decimal value of (A010051 interpreted as a binary number).
%C From Ferenc Adorjan (fadorjan(AT)freemail.hu): (Start)
%C Decimal expansion of the representation of the sequence of primes by a single real in (0,1).
%C 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.
%C Examples of the inverse mapping are A092855 and A092857. (End)
%C Is the prime constant an EL number? See Chow's 1999 article. - _Lorenzo Sauras Altuzarra_, Oct 05 2020
%C 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
%H Harry J. Smith, <a href="/A051006/b051006.txt">Table of n, a(n) for n = 0..20000</a>
%H Ferenc Adorjan, <a href="http://web.axelero.hu/fadorjan/aronsf.pdf">Binary mapping of monotonic sequences and the Aronson function</a>.
%H Timothy Y. Chow, <a href="http://timothychow.net/closedform.pdf">What is a Closed-Form Number?</a>, The American Mathematical Monthly, Vol. 106, No. 5. (May, 1999), pp. 440-448.
%H Igor Pak, <a href="https://arxiv.org/abs/1803.06636">Complexity problems in enumerative combinatorics</a>, arXiv:1803.06636 [math.CO], 2018.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=zoB8P-zOX_0">What is the prime constant and is it irrational?</a>, YouTube video, 2022.
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/primesbin.txt">Primes coded in binary to 1000 digits</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeConstant.html">Prime Constant</a>.
%F Prime constant C = Sum_{k>=1} 1/2^prime(k), where prime(k) is the k-th prime. - _Alexander Adamchuk_, Aug 22 2006
%F From _Amiram Eldar_, Aug 11 2020: (Start)
%F Equals Sum_{k>=1} A010051(k)/2^k.
%F Equals Sum_{k>=1} 1/A034785(k).
%F Equals (1/2) * A119523.
%F Equals Sum_{k>=1} pi(k)/2^(k+1), where pi(k) = A000720(k). (End)
%e 0.414682509851111660... (base 10) = .01101010001010001010001... (base 2).
%p 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
%t RealDigits[ FromDigits[ {{Table[ If[ PrimeQ[n], 1, 0], {n, 370}]}, 0}, 2], 10, 111][[1]] (* _Robert G. Wilson v_, Jan 15 2005 *)
%t RealDigits[Sum[1/2^Prime[k], {k, 1000}], 10, 100][[1]] (* _Alexander Adamchuk_, Aug 22 2006 *)
%o (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
%o (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
%o (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
%Y Cf. A000720, A010051, A034785, A051007, A132800, A092857, A092858, A092859, A092860, A092861, A092862, A092863, A092874, A119523, A370596.
%Y Cf. A036680 (Egyptian fractions).
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_
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