A243358 - OEIS

%I #46 Apr 03 2023 10:36:13

%S 2,2,2,3,5,7,11,19,37,83,223,739,3181,18911,166657,2375617,60916697,

%T 3199316947,403223394631,147983594957101,200280265936061027,

%U 1333721075205083093951,62146579709944366260614273,31146685223026045243771057244741

%N The densest possibly infinite sequence of primes of the form a(n) = floor[A^(C^n)] for A < 2. The density parameter C here approaches its minimal possible value C_0 = 1.2209864... (A117739), while the corresponding value of A is 1.8252076... (A243370).

%C Double-checked by David J. Broadhurst. Terms from a(61) to a(67) from David J. Broadhurst. Terms after a(52) are strong probable primes.

%C It is very likely, but not yet proved, that the sequence is infinite. However, it is clear that for density parameter C < C_0 = 1.2209864... (see A117739) such a sequence must contain nonprime terms.

%H Andrey V. Kulsha, <a href="/A243358/b243358.txt">Table of n, a(n) for n = 1..40</a>

%H Andrey V. Kulsha and David J. Broadhurst, <a href="http://www.primefan.ru/stuff/math/b243358.txt">Table of n, a(n) for n = 1..67</a>

%H Chris K. Caldwell, <a href="https://t5k.org/notes/proofs/A3n.html">A proof of a generalization of Mills' Theorem</a>

%F Once the terms up to the prime 223 are known, the following algorithm works:

%F 1. assign P:=(the largest prime currently in the sequence)

%F 2. assign k:=(the distance between 83 and P in the sequence)

%F 3. assign C:=(logP/log84)^(1/k)

%F 4. assign P:=P^C

%F 5. if floor[P] is prime, add it to the sequence and go to 4

%F 6. add nextprime[P] to the sequence and go to 1

%F That algorithm gives heuristically as many terms as needed because the increment of C at step 3 becomes so tiny that the values of 84^(C^n) for n < k don't jump over integers anymore (although there's no proof).

%F So we have a(n) = floor[(84-0)^(C_0^(n-10))], where C_0 = 1.2209864... (see A117739), and "84-0" notation means that when C approaches C_0 from above, the necessary value of A brings A^(C^10) to 84 from below.

%Y Cf. A060699, A117739, A243370.

%K nonn

%O 1,1

%A _Andrey V. Kulsha_, Jun 03 2014