A092307 -id:A092307 - cp's OEIS frontend

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.

3, 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 239, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1223, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2243, 2447

Offset: 1

Jianing Song, Apr 29 2018

nonn

a(n) exists for all n, which is easily shown by Dirichlet's theorem on arithmetic progressions.
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.
Clearly all safe primes are in this sequence, and all terms except a(2) = 5 are == 3 (mod 4).

			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.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005385, A079148, A092307, A119660.

A[1]:= 3: L:= 2:
for i from 2 to 100 do
  p:= nextprime(A[i-1]);
  while igcd(L, p-1) > 2 do p:= nextprime(p) od:
  A[i]:= p;
  L:= ilcm(L, p-1);
od:
seq(A[i],i=1..100); # Robert Israel, Apr 29 2018

Corrected by Robert Israel, Apr 29 2018

cp's OEIS Frontend

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.

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