A059847 - OEIS

%I #13 Feb 08 2025 04:51:28

%S 5,17,67,149,127,673,1151,541,1399,1973,2203,4201,2999,4861,5623,

%T 20477,10007,12889,25523,19661,34123,58889,25523,40693,40343,35729,

%U 106087,107441,34123,134581,302399,212777,259099,370373,156007,507289,371027

%N a(n) = 2*p + 2*n - 1, where p is the least prime such that next_prime(2*p) - 2*p = 2*n - 1.

%C Previous name was: a(n)=2*p+2n-1, the smallest prime q such that p=[q-(2n-1)]/2 is prime. A special generalization of safe primes: 1 is replaced with 2n-1.

%H Amiram Eldar, <a href="/A059847/b059847.txt">Table of n, a(n) for n = 1..201</a>

%F Min{p|p and q=(p-2n+1)/2, p and q are primes}.

%F a(n) = 2*A059846(n) + 2*n - 1. - _Amiram Eldar_, Feb 08 2025

%e For n = 8, 2*n-1 = 15, a(8) = 541 because (541-15)/2 = 263 is the corresponding generalized Sophie Germain prime and {541, 263} is the smallest pair belonging to 15.

%t a[n_] := Module[{p = 2, m = 2*n-1}, While[NextPrime[2*p] != 2*p + m, p = NextPrime[p]]; 2*p + m]; Array[a, 37] (* _Amiram Eldar_, Feb 08 2025 *)

%o (PARI) list(len) = {my(v = vector(len), c = 0, i, p = 2); while(c < len, i = (nextprime(2*p+1) - 2*p + 1)/2; if(i <= len && v[i] == 0, c++; v[i] = 2*p + 2*i - 1); p = nextprime(p+1)); v;} \\ _Amiram Eldar_, Feb 08 2025

%Y Cf. A005384, A005385, A059846.

%K nonn

%O 1,1

%A _Labos Elemer_, Feb 26 2001

%E New name from _Amiram Eldar_, Feb 08 2025