A323176 Prime numbers generated by the formula a(n) = round(c^((5/4)^n)), where c is the real constant given below.

113, 367, 1607, 10177, 102217, 1827697, 67201679, 6084503671, 1699344564793, 1940223714629437, 12877001925259260821, 771380135526168946568519, 722912215706743477640066820689, 21079337353575904691781436731789131951, 45166994522409258021988187061430676460306223027, 20822194129240450122637347266336444580153717439156314146339

Offset: 1

Author

Simon Plouffe, Jan 05 2019

Keywords

• nonn

Comments

The constant c is given in the article [Plouffe, 2018] with 2600 digits of precision.

Examples

			a(1) = round(c^((5/4)^1)) = round(112.69...) =  113,
a(2) = round(c^((5/4)^2)) = round(367.17...) =  367,
a(3) = round(c^((5/4)^3)) = round(1607.2...) = 1607, etc..
		

Links

• Simon Plouffe, A set of formulas for primes, arXiv:1901.01849 [math.NT], 2019.
• Simon Plouffe, The calculation of p(n) and pi(n), arXiv:2002.12137 [math.NT], 2020.

Programs

Maple

# Computes the values according to the formula, v = 43.804..., e = 5/4, m the
# number of terms. Returns the real and the rounded values (primes).
val := proc(s, e, m)
local ll, v, n, kk;
    v := s;
    ll := [];
    for n to m do
        v := v^e; ll := [op(ll), v]
    end do;
    return [ll, map(round, ll)]
end;

Formula

a(n) = round(c^((5/4)^n)), where c is a real constant starting 43.80468771580293481859664562569089495081037087137495184074061328752670419506...