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A371065 a(1)=2; for n > 1, a(n) is the least prime number p > a(n-1) such that p + 2^(n-1) is a prime number.

%I A371065 #40 Mar 31 2024 02:07:44
%S A371065 2,3,7,11,13,29,37,53,61,89,127,131,157,197,223,269,307,359,367,419,
%T A371065 463,491,547,593,607,641,643,701,823,947,1213,1229,1237,1319,1327,
%U A371065 1451,1723,2381,3019,3299,3307,3371,3847,4493,4621,4931,5179,5783,6043,6197,6469
%N A371065 a(1)=2; for n > 1, a(n) is the least prime number p > a(n-1) such that p + 2^(n-1) is a prime number.
%H A371065 Amiram Eldar, <a href="/A371065/b371065.txt">Table of n, a(n) for n = 1..5000</a>
%e A371065 For n=5, the preceding term a(4)=11 and 2^(5-1)=16, so a(5) is the least prime p > 11 such that p+16 is a prime too, which is p = 13 = a(5).
%e A371065 From _Michael De Vlieger_, Mar 10 2024: (Start)
%e A371065 Table of first terms:
%e A371065    n   a(n)  2^(n+1)  a(n)+2^(n+1)
%e A371065   -------------------------------
%e A371065    1      2       1         3
%e A371065    2      3       2         5
%e A371065    3      7       4        11
%e A371065    4     11       8        19
%e A371065    5     13      16        29
%e A371065    6     29      32        61
%e A371065    7     37      64       101
%e A371065    8     53     128       181
%e A371065    9     61     256       317
%e A371065   10     89     512       601
%e A371065   11    127    1024      1151
%e A371065   12    131    2048      2179
%e A371065   ... (End)
%t A371065 a[1] = 2; a[n_] := a[n] = Module[{p = NextPrime[a[n - 1]]}, While[! PrimeQ[p + 2^(n - 1)], p = NextPrime[p]]; p]; Array[a, 50] (* _Amiram Eldar_, Mar 10 2024 *)
%Y A371065 Cf. A108184, A056206.
%K A371065 nonn
%O A371065 1,1
%A A371065 _Ahmad J. Masad_, Mar 09 2024