A348019 - OEIS

%I #6 Sep 24 2021 16:34:47

%S 2,0,3,0,0,0,5,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Decimal expansion of Sum_{k>=1} prime(k)/10^(2^k).

%C This constant appears in a formula derived by Sierpiński (1952) that generates all the prime numbers.

%C As in the case of other formulas for calculating prime numbers, this formula is not practical: for calculating the k-th prime one has to calculate the first k primes in order to get the constant with enough accuracy.

%D Ralph G. Archibald, An introduction to the theory of numbers, Merrill, 1970, p. 282.

%D Raymond Ayoub, An introduction to the analytic theory of numbers, AMS, 1963, p. 129.

%D Paulo Ribenboim, The New Book of Prime Number Records, Springer, 1996, p. 182.

%H Chris K. Caldwell, <a href="https://primes.utm.edu/notes/faq/p_n.html">FAQ: Is there a formula for the nth Prime?</a>, The PrimePages.

%H Paul Pollack, <a href="http://pollack.uga.edu/NABDofficial.pdf">Not Always Buried Deep: A Second Course in Elementary Number Theory, AMS, 2009, p. 13.

%H Andrzej Rotkiewicz, <a href="https://doi.org/10.1007/BF02844227">W. Sierpiński’s works on the Theory of Numbers</a>, Rend. Circ. Mat. Palermo, Vol. 21 (1972), pp. 5-24.

%H Wacław Sierpiński, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k31876/f1078.item">Sur une formule donnant tous les nombres premiers</a>, Comptes rendus hebdomadaires des séances de l'Académie des sciences, Vol. 235, No. 19 (1952), pp. 1078-1079; <a href="https://archive.org/details/ComptesRendusAcademieDesSciences0235/page/n1078/mode/2up">alternative link</a>.

%F prime(n) = floor(c * 10^(2^n)) - 10^(2^(n-1)) * floor(c * 10^(2^(n-1))), where c is this constant (Sierpiński, 1952).

%e 0.02030005000000070000000000000011000000000000000000...

%t RealDigits[Sum[Prime[i]/10^(2^i), {i, 1, 7}]][[1]]

%Y Similar constants: A051021, A079614, A086238, A300753.

%K nonn,cons,easy

%O -1,1

%A _Amiram Eldar_, Sep 24 2021