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A303705 a(1) = 3; a(n) is the smallest prime such that gcd(a(i)-1, a(n)-1) = 2 holds for 1 <= i < n.

%I A303705 #20 May 26 2019 19:38:04
%S A303705 3,5,7,11,23,47,59,83,107,167,179,227,239,263,347,359,383,467,479,503,
%T A303705 563,587,719,839,863,887,983,1019,1187,1223,1283,1307,1319,1367,1439,
%U A303705 1487,1523,1619,1823,1907,2027,2039,2063,2099,2207,2243,2447
%N A303705 a(1) = 3; a(n) is the smallest prime such that gcd(a(i)-1, a(n)-1) = 2 holds for 1 <= i < n.
%C A303705 a(n) exists for all n, which is easily shown by Dirichlet's theorem on arithmetic progressions.
%C A303705 Apart from 3, the first term that is not a term in A005385 is 239. The first term in A092307 and A119660 (apart from 2) that is not a term here is 443.
%C A303705 Clearly all safe primes are in this sequence, and all terms except a(2) = 5 are == 3 (mod 4).
%H A303705 Robert Israel, <a href="/A303705/b303705.txt">Table of n, a(n) for n = 1..10000</a>
%e A303705 a(13) = 239 since lcm(a(1)-1, a(2)-1, ..., a(12)-1) = 2^2*3*5*11*23*29*41*53*83*89*113 and 239-1 = 2*7*17.
%p A303705 A[1]:= 3: L:= 2:
%p A303705 for i from 2 to 100 do
%p A303705   p:= nextprime(A[i-1]);
%p A303705   while igcd(L, p-1) > 2 do p:= nextprime(p) od:
%p A303705   A[i]:= p;
%p A303705   L:= ilcm(L, p-1);
%p A303705 od:
%p A303705 seq(A[i],i=1..100); # _Robert Israel_, Apr 29 2018
%Y A303705 Cf. A005385, A079148, A092307, A119660.
%K A303705 nonn
%O A303705 1,1
%A A303705 _Jianing Song_, Apr 29 2018
%E A303705 Corrected by _Robert Israel_, Apr 29 2018