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A075712 Rearrangement of primes into Germain groups (or Cunningham chains).

%I A075712 #22 Nov 14 2021 01:21:48
%S A075712 2,5,11,23,47,3,7,13,17,19,29,59,31,37,41,83,167,43,53,107,61,67,71,
%T A075712 73,79,89,179,359,719,1439,2879,97,101,103,109,113,227,127,131,263,
%U A075712 137,139,149,151,157,163,173,347,181,191,383,193,197,199,211,223,229,233
%N A075712 Rearrangement of primes into Germain groups (or Cunningham chains).
%C A075712 In each group, p(i+1) = 2*p(i)+1.
%C A075712 The groups are also known as Cunningham chains of the first kind.
%H A075712 Michael De Vlieger, <a href="/A075712/b075712.txt">Table of n, a(n) for n = 1..10000</a>
%e A075712 The groups are:
%e A075712 {2, 5, 11, 23, 47},
%e A075712 {3, 7},
%e A075712 {13},
%e A075712 {17},
%e A075712 {19},
%e A075712 {29, 59},
%e A075712 {31},
%e A075712 {37},
%e A075712 {41, 83, 167},
%e A075712 {43},
%e A075712 {53, 107},
%e A075712 {61},
%e A075712 {67},
%e A075712 {71},
%e A075712 {73},
%e A075712 {79},
%e A075712 {89, 179, 359, 719, 1439, 2879},
%e A075712 {97},
%e A075712 {101},
%e A075712 {103},
%e A075712 {109},
%e A075712 {113, 227},
%e A075712 {127},
%e A075712 {131, 263},
%e A075712 {137},
%e A075712 {139},
%e A075712 ...
%t A075712 Block[{a = {2}, j = 1, k, p}, Do[k = j; If[PrimeQ@ a[[-1]], AppendTo[a, 2 a[[-1]] + 1], While[! FreeQ[a, Set[p, Prime[k]]], k++]; j++; Set[a, Append[a[[1 ;; -2]], p]]], 10^3]; a] (* _Michael De Vlieger_, Nov 17 2020 *)
%o A075712 (PARI) first(n) = my(res=List([2,5,11,23,47])); forprime(p=3, oo, if(!isprime((p-1)>>1), listput(res,p); c = 2*p+1; while(isprime(c), listput(res,c); c=2*c+1)); if(#res>n,return(res))); res \\ _David A. Corneth_, Nov 13 2021
%Y A075712 Cf. A005384, A059452, A059453, A059455, A059456, A053176.
%Y A075712 See also A181697.
%Y A075712 See A059456 for initial terms, A338945 for lengths.
%K A075712 nonn,tabf
%O A075712 1,1
%A A075712 _Zak Seidov_, Oct 03 2002
%E A075712 Edited by _N. J. A. Sloane_, Nov 13 2021
%E A075712 More terms from _David A. Corneth_, Nov 13 2021