A348019 Decimal expansion of Sum_{k>=1} prime(k)/10^(2^k).

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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

-1,1

Comments

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

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.

References

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

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

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

Links

Table of n, a(n) for n=-1..85.

Chris K. Caldwell, FAQ: Is there a formula for the nth Prime?, The PrimePages.

Paul Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, AMS, 2009, p. 13.

Andrzej Rotkiewicz, W. Sierpiński’s works on the Theory of Numbers, Rend. Circ. Mat. Palermo, Vol. 21 (1972), pp. 5-24.

Wacław Sierpiński, Sur une formule donnant tous les nombres premiers, Comptes rendus hebdomadaires des séances de l'Académie des sciences, Vol. 235, No. 19 (1952), pp. 1078-1079; alternative link.

Formula

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

Example

0.02030005000000070000000000000011000000000000000000...

Mathematica

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

Crossrefs

Similar constants: A051021, A079614, A086238, A300753.

Sequence in context: A348223 A035199 A035229 * A285982 A261727 A234579

Adjacent sequences: A348016 A348017 A348018 * A348020 A348021 A348022

Keyword

nonn,cons,easy

Author

Amiram Eldar, Sep 24 2021

Status

approved