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Decimal expansion of Varona's constant = Sum_{k >= 1} prime(k)/2^(k + k!).
1

%I #27 Jan 14 2023 13:36:57

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

%T 0,0,2,5,8,6,0,8,7,5,7,1,8,0,9,0,3,2,5,1,7,5,3,1,2,2,0,3,8,1,8,8,8,9,

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

%N Decimal expansion of Varona's constant = Sum_{k >= 1} prime(k)/2^(k + k!).

%C Varona's constant v is transcendental and generates the primes via prime(1)=2=floor(4*v) and for n>1 prime(n) = floor(v*2^(n+n!)) - 2^(1+n!-(n-1)!)*floor(v*2^(n-1+(n-1)!)).

%H Juan L. Varona, <a href="https://arxiv.org/abs/2012.11750">A couple of transcendental prime-representing constants</a>, arXiv:2012.11750 [math.NT], 2020.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 0.69726565107703208923339843750000000025860875718090325175312203818889

%t First@RealDigits@N[Sum[Prime[i]/2^(i + i!), {i, 1, 12}], 300]

%o (PARI) suminf(k=1, prime(k)/2^(k + k!)) \\ _Michel Marcus_, Dec 21 2020

%Y Cf. A249270, A016104, A051021.

%K nonn,cons

%O 0,1

%A _José María Grau Ribas_, Dec 20 2020