Primes Generated by Alternating Factorials?

  Found this one in an article by Richard Guy on the Strong Law of Small Numbers... *K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

or with the recurrence relation

${\displaystyle \mathrm{af}(𝑛)=𝑛!-\mathrm{af}(𝑛-1)}$

in which af(1) = 1.  

{Just a quick note on the notation af(n) for alternating factorial.  I would think something like +/- n! is much more intuitive.}

Is it always true?  Is even the next one true?

Just a quick note on the notation a(n) for alternating factorial.  I would think something like +/- n! is much more intuitive.

Spoiler (of sorts)

x
x
x
xx

xx
x
xx
x
x
x
Neil Calkin ‏@neil_calkin offers:
solutions for 3,4,5,6,7,8,10,15,19,41,59,61,105,160 no more small values.  

It continues 661, 2653, 3069, and probables (3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961) ...  but the sequence is finite, and all prime terms in alternating factorials must be less than p = 3612701,

Only the values up to n = 661 had been proved prime in 2006. af(661) is approximately 7.818097272875 × 101578.

If you love factor challenges  f(9) = 36614981, and f(11)=36614981  (maybe easier)

 

The prime values of the prime terms are 

3- 5

4 19

5 101

6- 619

7- 4421
8- 35899
10- 3301819
15- 1226280710981
19 115578717622022981
41- 32656499591185747972776747396512425885838364422981
59-136372385605079432248118270297843987319730859689490659519593045108637838364422981

61-499395599150088488088828589263699706832570087241364247806476254829684637838364422981
105-1071195818389184106041377222623114315174404652995290026861977169467051355218307761044337430404771512503239158647256903838408052353602736923780521178553460637838364422981
160-468544077492065936712052044718939948687543330546977719976017418129955876663406131164377030450551575840099843957105136480237871017419158043635450756712088769133544426722033165168878328322819566779381528981882285541609256481166622331374702000809600061055686236758821446539362161635577019