[qnvttpzl] Cunningham chains in recursive prime predecessor factorization notation

Here are the ending primes of some Cunningham chains of the first kind written in compressed recursive prime predecessor factorization notation (RPPFN). In a nutshell, value[(X Y)] is 1+2*value[X]*value[Y]. The notable feature is the long string of initial open parentheses. We continue to use this Pari/GP code. We'll leave the trailing parentheses because they look impressive. First we do OEIS sequence A005602.

a=[13, 3, 41, 509, 2, 89, 1122659, 19099919, 85864769, 26089808579, 665043081119, 554688278429, 4090932431513069, 95405042230542329]
embiggen(x)=while(isprime(2*x+1),x=2*x+1);x
for(i=1,14,b=embiggen(a[i]);print(b," ");f(b);print(""))

f(13) = (2())
f(7) = (())
f(167) = (((22(2))))
f(4079) = ((((2(() () (()))))))
f(47) = ((((2))))
f(2879) = ((((((22((2))))))))
f(71850239) = (((((((((22(2))) (() (()) (()) (((2)))))))))))
f(2444789759) = (((((((((2() ()) ((((2))) (() (2)) (2() () (2)))))))))))
f(21981381119) = (((((((((22222((2)) ((2()) ((2) (()) (() ((2))))))))))))))
f(13357981992959) = (((((((((((2() () ()) (2(2) (()) ((2)) (222() ((22(2(2) (2))))))))))))))))
f(681004115066879) = ((((((((((((2()) ((2(2) (2()) ((22(2(())))) (() (2) (() () () (2) (2()))))))))))))))))
f(1136001594224639) = ((((((((((((2(() () () (()) (((2)) (2() (() () (2(()))) (2222((22(2)))))))))))))))))))
f(16756459239477534719) = (((((((((((((2(() () (2(())) (2() () (2222(2) (2) (2))) (2() ((2)) (22((2)) (() () (2) (2()))))))))))))))))))
f(781558105952602767359) = ((((((((((((((22((2) (())) ((22(2))) (((2)) (() (2(())) (22(2)) (() (()) (() () (2(()))) ((2(22() ()))))))))))))))))))))

The number of initial parentheses does not grow by exactly one each entry because sometimes the RPPFN of the initial point also begins with some parentheses.

CC15 113220800675069784839 found by Carmody may not be minimal ("Sophie Germain tries to catch Cunningham") .
f(1855009598260343354818559) = ((((((((((((((((() ()) (22() (2) (2)) (() (222(222() (())))) ((222() () (2())) (((2) (2) (() (2)) (() (2())) (((22(2))))))))))))))))))))))

Here are a few more from Dirk Agustin's Cunningham Chain records page.

CC16 810433818265726529159
f(26556295356931326907514879) = (((((((((((((((((() (((2)))) ((2) (2) (222(2) (2))) (22(2) (2) (()) (222)) ((2) (222() (2) (()) (((2))) (22222(2()) (() ()))))))))))))))))))))

CC17 2759832934171386593519
f(180868411173855991792926719) = ((((((((((((((((((2222()) (222() (2(())) ((2) (() ()))) (() (2) (2() (222) (22() (2())))) (2222(()) (() ()) ((222) (22222()))))))))))))))))))))