A394683 a(n) = (m*p)^2 + 1 where p = A002496(n) is the n-th prime of the form m^2 + 1.

5, 101, 4625, 49285, 1020101, 7606565, 16908545, 64320401, 191767105, 309830405, 2180142865, 4101121601, 24811920325, 30860651585, 82691903845, 164266468805, 351397612945, 531572228101, 690025939685, 1771853832101, 2436758464145, 2986398734401, 3635687935505, 4002008252005, 4827380236901

Offset

1,1

Comments

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.

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.

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

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.

Links

Table of n, a(n) for n=1..25.

Formula

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

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

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

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

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

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

Example

a(1) = (1*2)^2 + 1 = 5, since 2 = 1^2 + 1 is the first Landau prime.
a(2) = (2*5)^2 + 1 = 101, since 5 = 2^2 + 1.
a(3) = (4*17)^2 + 1 = 4625 = 5^3 * 37, since 17 = 4^2 + 1.
a(4) = (6*37)^2 + 1 = 49285 = 5 * 9857, since 37 = 6^2 + 1.
a(5) = (10*101)^2 + 1 = 1020101 (prime), since 101 = 10^2 + 1.

Prog

(PARI) for(m=1, 200, my(p=m^2+1); if(isprime(p), print1((m*p)^2+1, ", ")))
(Python) from sympy import isprime; [((m*(m*m+1))**2+1) for m in range(1, 200) if isprime(m*m+1)]

Crossrefs

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

Sequence in context: A074790 A009757 A119556 * A009765 A113073 A349045

Adjacent sequences: A394680 A394681 A394682 * A394684 A394685 A394686

Keyword

nonn,easy

Author

Steven Lee Benjamin, Apr 27 2026

Status

approved