A352959 a(n) is the first prime to start a sequence of exactly n primes under the iteration p -> (p^2 + 3*p + 1)/5.

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

Offset

1,1

Comments

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.

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

a(8) > 5*10^14 - Bert Dobbelaere, Apr 17 2022

Links

Table of n, a(n) for n=1..7.

Example

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.

Maple

g:= proc(x) if isprime(x) then 1+procname((x^2+3*x+1)/5) else 0 fi end proc:
V:= Vector(5): V[1]:=2: count:= 1:
for x from 11 by 10 while count < 5 do
  v:= g(x); if v>0 and V[v] = 0 then count:= count+1; V[v]:= x; fi;
od:
convert(V, list);

Mathematica

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 *)

Crossrefs

Cf. A352954.

Sequence in context: A377788 A178194 A069460 * A117816 A237263 A099189

Adjacent sequences: A352956 A352957 A352958 * A352960 A352961 A352962

Keyword

nonn,more

Author

J. M. Bergot and Robert Israel, Apr 11 2022

Extensions

a(6) from Amiram Eldar, Apr 11 2022

a(7) from Bert Dobbelaere, Apr 17 2022

Status

approved