Theorem: Smallest Cunningham Chain of the Second Kind of Length 13

The smallest Cunningham chain of the second kind of length $13$ is:
:$758 \, 083 \, 947 \, 856 \, 951$, $1 \, 516 \, 167 \, 895 \, 713 \, 901$, $3 \, 032 \, 335 \, 791 \, 427 \, 801$, $6 \, 064 \, 671 \, 582 \, 855 \, 601$, $12 \, 129 \, 343 \, 165 \, 711 \, 201$, $24 \, 258 \, 686 \, 331 \, 422 \, 401$, $48 \, 517 \, 372 \, 662 \, 844 \, 801$, $97 \, 034 \, 745 \, 325 \, 689 \, 601$, $194 \, 069 \, 490 \, 651 \, 379 \, 201$, $388 \, 138 \, 981 \, 302 \, 758 \, 401$, $776 \, 277 \, 962 \, 605 \, 516 \, 801$, $1 \, 552 \, 555 \, 925 \, 211 \, 033 \, 601$, $3 \, 105 \, 111 \, 850 \, 422 \, 067 \, 201$

Proof:
Let $C$ denote the sequence in question.
We have that $758 \, 083 \, 947 \, 856 \, 951$ is prime.
First note that:
:$\dfrac {758 \, 083 \, 947 \, 856 \, 951 + 1} 2 = 379 \, 041 \, 973 \, 928 \, 476 = 2^2 \times 94 \, 760 \, 493 \, 482 \, 119$
and so is not prime.
Thus $758 \, 083 \, 947 \, 856 \, 951$ fulfils the requirement for $C$ to be a Cunningham chain of the second kind.
Then:
{{begin-eqn}}
{{eqn | n = 1
      | l = 2 \times 758 \, 083 \, 947 \, 856 \, 951 - 1
      | r = 1 \, 516 \, 167 \, 895 \, 713 \, 901
      | c = which is prime
}}
{{eqn | n = 2
      | l = 2 \times 1 \, 516 \, 167 \, 895 \, 713 \, 901 - 1
      | r = 3 \, 032 \, 335 \, 791 \, 427 \, 801
      | c = which is prime
}}
{{eqn | n = 3
      | l = 2 \times 3 \, 032 \, 335 \, 791 \, 427 \, 801 - 1
      | r = 6 \, 064 \, 671 \, 582 \, 855 \, 601
      | c = which is prime
}}
{{eqn | n = 4
      | l = 2 \times 6 \, 064 \, 671 \, 582 \, 855 \, 601 - 1
      | r = 12 \, 129 \, 343 \, 165 \, 711 \, 201
      | c = which is prime
}}
{{eqn | n = 5
      | l = 2 \times 12 \, 129 \, 343 \, 165 \, 711 \, 201 - 1
      | r = 24 \, 258 \, 686 \, 331 \, 422 \, 401
      | c = which is prime
}}
{{eqn | n = 6
      | l = 2 \times 24 \, 258 \, 686 \, 331 \, 422 \, 401 - 1
      | r = 48 \, 517 \, 372 \, 662 \, 844 \, 801
      | c = which is prime
}}
{{eqn | n = 7
      | l = 2 \times 48 \, 517 \, 372 \, 662 \, 844 \, 801 - 1
      | r = 97 \, 034 \, 745 \, 325 \, 689 \, 601
      | c = which is prime
}}
{{eqn | n = 8
      | l = 2 \times 97 \, 034 \, 745 \, 325 \, 689 \, 601 - 1
      | r = 194 \, 069 \, 490 \, 651 \, 379 \, 201
      | c = which is prime
}}
{{eqn | n = 9
      | l = 2 \times 194 \, 069 \, 490 \, 651 \, 379 \, 201 - 1
      | r = 388 \, 138 \, 981 \, 302 \, 758 \, 401
      | c = which is prime
}}
{{eqn | n = 10
      | l = 2 \times 388 \, 138 \, 981 \, 302 \, 758 \, 401 - 1
      | r = 776 \, 277 \, 962 \, 605 \, 516 \, 801
      | c = which is prime
}}
{{eqn | n = 11
      | l = 2 \times 776 \, 277 \, 962 \, 605 \, 516 \, 801 - 1
      | r = 1 \, 552 \, 555 \, 925 \, 211 \, 033 \, 601
      | c = which is prime
}}
{{eqn | n = 12
      | l = 2 \times 1 \, 552 \, 555 \, 925 \, 211 \, 033 \, 601 - 1
      | r = 3 \, 105 \, 111 \, 850 \, 422 \, 067 \, 201
      | c = which is prime
}}
{{eqn | n = 13
      | l = 2 \times 3 \, 105 \, 111 \, 850 \, 422 \, 067 \, 201 - 1
      | r = 6 \, 210 \, 223 \, 700 \, 844 \, 134 \, 401
      | c = which is $41 \times 691 \times 739 \times 4957 \times 59 \, 838 \, 677$ and so not prime
}}
{{end-eqn}}
Establishing that this is indeed the smallest such Cunningham chain of the second kind of length $13$ can be done by a computer search.
{{qed}}