[nmrqcxez] Finite sequences of primes

Fermat numbers grow as O(2^2^n).  Apply the Prime Number Theorem as a heuristic -- a number n is prime with probability 1/log(n) -- and then replace the sum with an integral, then conclude that there are only a finite expected number of Fermat primes.  A similar approximation argument works for any sequence growing as (1+alpha)^(1+beta)^n, for positive alpha and beta.

Here are some more slowly growing asymptotic sequence growth rates that would also have a finite expected number of primes.  Fermat numbers grow so fast that they are overkill.

(1+alpha)^n^(1+beta)

(1+alpha)^(n*(log n)^(1+beta))

(1+alpha)^(n*(log n)*(log log n)^(1+beta))

These follow from the convergence of the integrals of 1/x^(1+beta), 1/(x*(log x)^(1+beta)), and 1/(x*(log x)*(log log x)^(1+beta)).

Incidentally, Mersenne numbers grow as 2^(n*log n), so we expect an infinite number of them to be prime, but only barely.  The integral of 1/(x*log x) is log log x, which diverges extremely slowly.

Are there any sequences that grow faster than (1+alpha)^(n*(log n)*(log log n)*(log log log n)*...) that have an infinite number of primes?  We probably need to limit the type of sequences somehow, or else it would be easy to construct something along the lines of f(n)=nextprime(2^2^n).

We explored 1.5^n^1.5 (n <= 3820) and found the following primes:

ceil(1.5^1^1.5) = 2
floor(1.5^2^1.5) = 3
floor(1.5^8^1.5) = 9649
ceil(1.5^42^1.5) = 852069747811116972109875222464252771091039570931
ceil(1.5^60^1.5) = 6915469149559541382089000664679518689857578042557766452578156297444346096339139131
floor(1.5^61^1.5) = 784004647503044231367089868577092132418449967431905553965609437027199675493821852237
ceil(1.5^163^1.5) = 2843017533260985010694548776892309855603589981873575072997240935668178294887672889892483420546588216491406511265136751706516520216010851901076208798125523476834397113088228375446152254181894670397711424402046795371831348295827170539437281033397169434713977200649666292247119320731085463593586047121077004471593138233344780787933699241119783143640475074753926994619789

We explored 1.5^(n*log(n)^1.5) (n <= 6846) and found the following primes:

ceil(1.5^(2*log(2)^1.5)) = 2
ceil(1.5^(3*log(3)^1.5)) = 5
ceil(1.5^(8*log(8)^1.5)) = 16759
floor(1.5^(50*log(50)^1.5)) = 133515421876799711956512714908896295020000818230293796389265675254297
floor(1.5^(73*log(73)^1.5)) = 1735871086055272518034305068724289808613087895373465352838436980978395419910575340052199712423367433395137993242363

Next, we explore sequences which much closer to the boundary between expecting a finite versus infinite number of primes.  (Possibly previously related: What is the derivative of iterated logarithm?)  We define the product of iterated natural logarithm function in Pari/GP as follows:

productiteratedlog(x)=local(y);y=log(x);if(y>1,x*productiteratedlog(y),x)

The integral of the reciprocal of this function is a piecewise thing that behaves like log x + C for a while, then log log x + C for a while, then log log log x + C, and so on.  It therefore diverges (so these sequences are expected to contain an infinite number of primes).  Can we define a function whose integral diverges even more slowly?  (What's the derivative of the inverse Ackerman function?)

We explored sequences of the form exp(productiteratedlog(n)) up to n=4811 and found the following primes.  Note that there are no primes of the form exp(n*log(n)) because that is equivalent to the n^n.

floor(exp(1)) = 2
ceil(exp(1)) = 3
floor(exp(2)) = 7
floor(exp(30*log(30)*log(log(30)))) = 1760128056820064244010305004494635245456824791132541637

We explored sequences of the form 1.5^productiteratedlog(n) up to n=8587 and found the following primes.  Note that "log" is still the natural logarithm function, not log base 1.5.

ceil(1.5^1) = 2
floor(1.5^2) = 2
ceil(1.5^2) = 3
floor(1.5^(3*log(3))) = 3
ceil(1.5^(6*log(6))) = 79
ceil(1.5^(7*log(7))) = 251
floor(1.5^(11*log(11))) = 44129
floor(1.5^(13*log(13))) = 744127
floor(1.5^(18*log(18)*log(log(18)))) = 5294466019
ceil(1.5^(49*log(49)*log(log(49)))) = 4282685512139315751672739845547275333810677903
ceil(1.5^(139*log(139)*log(log(139)))) = 6221632666571163256101566337246863650786499780605768974125245025330883899622185682413811743744661868380530596471794351854330573314331143314872956807522819830556618344028551855268029363025956851
ceil(1.5^(211*log(211)*log(log(211)))) = 3623494910954830585594313737287881530765752272229835760527651812393582429644929820469389670214667905333199180860386175763265704984813233955603461933338053203761636195958993503806976854699767461110674905487021598284019232000497521739654465437636807465852318560446512975205883072019435693838691994972399554208850816983281868149450551863
floor(1.5^(352*log(352)*log(log(352)))) = 7289878734250178733557869371226896257197695691444233340924792985960426450348980980173587882815055284500903729584130609072757886746227645416034122373248464805270371112957337295293685646178665534636450775554372470252121103912396741477323903733624670105182142642422668301774110539909227413055340142789999395433643294917957997726155270670994619845935016902355835220563425700321313729883919366920336703580184344433746670937276952428746193471967580404410209867212289698162796403647146180978879155890648375724608924682947343951640209056626802452338726872019682305806205921186649908142725637212012518872063615446327545327031106057072327429927943759967
ceil(1.5^(2704*log(2704)*log(log(2704)))) ~= 2.113 * 10^7788