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A348019
Decimal expansion of Sum_{k>=1} prime(k)/10^(2^k).
0
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
Chris K. Caldwell, FAQ: Is there a formula for the nth Prime?, The PrimePages.
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
KEYWORD
nonn,cons,easy
AUTHOR
Amiram Eldar, Sep 24 2021
STATUS
approved