A394683 - OEIS

%I #15 May 02 2026 18:56:38

%S 5,101,4625,49285,1020101,7606565,16908545,64320401,191767105,

%T 309830405,2180142865,4101121601,24811920325,30860651585,82691903845,

%U 164266468805,351397612945,531572228101,690025939685,1771853832101,2436758464145,2986398734401,3635687935505,4002008252005,4827380236901

%N a(n) = (m*p)^2 + 1 where p = A002496(n) is the n-th prime of the form m^2 + 1.

%C a(n) is the first iterate of the Gaussian norm dynamics: if N_0(m) = m^2 + 1, then a(n) = N_1(m) = (m*(m^2+1))^2 + 1 at values m where N_0(m) is prime.

%C a(n) is the Gaussian norm of the integer m*p + i in Z[i], where p = m^2 + 1 is the n-th Landau prime.

%C gcd(a(n), A002496(n)) = 1 for all n, and a(n) == 1 (mod A002496(n)^2).

%C The tower of iterates producing a(n) is governed by the cubic field Q(alpha), alpha^3 = alpha + 1, whose Artin L-function has Dirichlet coefficients A030199.

%F a(n) = A002496(n)^3 - A002496(n)^2 + 1.

%F a(n) = A005574(n)^2 * A002496(n)^2 + 1.

%F a(n) = A382617(n)^2 + 1.

%F a(n) = A100104(A002496(n)).

%F a(n) = A078429(A002496(n)^3).

%F a(n) = Phi(A005574(n))^2 + 1 where Phi(m) = m^3 + m.

%e a(1) = (1*2)^2 + 1 = 5, since 2 = 1^2 + 1 is the first Landau prime.

%e a(2) = (2*5)^2 + 1 = 101, since 5 = 2^2 + 1.

%e a(3) = (4*17)^2 + 1 = 4625 = 5^3 * 37, since 17 = 4^2 + 1.

%e a(4) = (6*37)^2 + 1 = 49285 = 5 * 9857, since 37 = 6^2 + 1.

%e a(5) = (10*101)^2 + 1 = 1020101 (prime), since 101 = 10^2 + 1.

%o (PARI) for(m=1, 200, my(p=m^2+1); if(isprime(p), print1((m*p)^2+1, ", ")))

%o (Python) from sympy import isprime; [((m*(m*m+1))**2+1) for m in range(1,200) if isprime(m*m+1)]

%Y Cf. A002496, A005574, A030199, A078429, A100104, A382617.

%K nonn,easy

%O 1,1

%A _Steven Lee Benjamin_, Apr 27 2026