A352959 - OEIS

%I #24 Apr 17 2022 03:46:09

%S 2,31,11,271,38547571,11480934901,801479967311

%N a(n) is the first prime to start a sequence of exactly n primes under the iteration p -> (p^2 + 3*p + 1)/5.

%C If x_0 = a(n) and x_{k+1} = (x_k^2 + 3*x_k + 1)/5, then x_0, x_1, ..., x_{n-1} are prime but x_n is not prime.

%C For n > 1, a(n) == 1 (mod 10).

%C a(8) > 5*10^14 - _Bert Dobbelaere_, Apr 17 2022

%e a(3) = 11 because 11, (11^2 + 3*11 + 1)/5 = 31 and (31^2 + 3*31 + 1)/5 = 211 are prime but (211^2 + 3*211 + 1)/5 = 9031 is composite.

%p g:= proc(x) if isprime(x) then 1+procname((x^2+3*x+1)/5) else 0 fi end proc:

%p V:= Vector(5): V[1]:=2: count:= 1:

%p for x from 11 by 10 while count < 5 do

%p   v:= g(x); if v>0 and V[v] = 0 then count:= count+1; V[v]:= x; fi;

%p od:

%p convert(V,list);

%t f[p_] := -1 + Length @ NestWhileList[(#^2 + 3*# + 1)/5 &, p, PrimeQ]; seq[len_, nmax_] := Module[{s = Table[0, {len}], p = 1, c = 0, i}, While[c < len && p < nmax, p = NextPrime[p]; i = f[p]; If[i <= len && s[[i]] == 0, c++; s[[i]] = p]]; s]; seq[5, 10^8] (* _Amiram Eldar_, Apr 11 2022 *)

%Y Cf. A352954.

%K nonn,more

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Apr 11 2022

%E a(6) from _Amiram Eldar_, Apr 11 2022

%E a(7) from _Bert Dobbelaere_, Apr 17 2022