

A108739


Mills' constant A generates a sequence of primes via b(n)= floor(A^3^n). This sequence is a(n) = b(n+1)b(n)^3.


6



3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, 66768, 300840, 1623568
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

This allows larger terms of A051254 (which triple in digits each entry) to be given. Like A051254, currently requires Riemann Hypothesis to show sequence continues.
Likewise a(12) and a(13) generate only a probable prime numbers, as well as being conditional on a(11) and a(12) being proved primes. Minimality of a(12)a(13) is exhaustively tested.  Serge Batalov, Aug 06 2013


REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 8.


LINKS



FORMULA

b(1) = 2; b(n+1) = nextprime(b(n)^3); a(n) = b(n+1)b(n)^3;


EXAMPLE

The Mills' primes (given in A051254) are 2, 2^3+3 = 11, (2^3+3)^3+30 = 11^3+30 = 1361, ((2^3+3)^3+30)^3+6 = 1361^3+6 = 2521008887, etc. The terms added at each step yield this sequence. They are the least positive integers which added to the cube of the preceding prime yield again a prime, cf. formula.  M. F. Hasler, Jul 22 2013


MATHEMATICA

B[1] = 2; B[n_] := B[n] = NextPrime[B[n  1]^3]; Table[B[n + 1]  B[n]^3, {n, 7}] (* Robert Price, Jun 09 2019 *)


PROG

(PARI) p=2; until(, np=nextprime(p^3); print1(npp^3, ", "); p=np) \\ Jeppe Stig Nielsen, Apr 22 2020


CROSSREFS



KEYWORD

more,nonn,hard


AUTHOR



EXTENSIONS

a(9)a(11) from Caldwell and Cheng, Aug 29 2005


STATUS

approved



