A392244 - OEIS

%I #30 Jan 12 2026 15:59:48

%S 1,6,36,225,1645,12706,104894,892723,7755330,68588950,614983774,

%T 5573589175

%N Number of primes of the form b^2 + (b+1)^2 for b <= 10^n.

%C Conjecture 1: Lim_{n->oo} a(n)/A206709(n) = 2  (where A206709(n) = pi_{b^2+1}(10^n)).

%C == ===============

%C  n a(n)/A206709(n)

%C == ===============

%C  0 1

%C  1 1.2

%C  2 1.894737

%C  3 2.008929

%C  4 1.956005

%C  5 1.908954

%C  6 1.938537

%C  7 1.956173

%C  8 1.961299

%C  9 1.965287

%C 10 1.968843

%C 11 1.97178

%C .

%C Conjecture 2 (conjecture 1 + A206709 conjecture):

%C   Lim_{n->oo} a(n)/A006880(n) = 1.372826  (where A006880(n) = pi(10^n)).

%C A proof that a(n) is a majorant of A006880(n) would give a nonlinear function with infinitely many primes.

%F a(n) = pi_{b^2+(b+1)^2}(10^n).

%e a(1) = 6 because there are 6 primes of the form b^2 + (b+1)^2 for b <= 10^1: 1, 2, 4, 5, 7, 9.

%o (PARI) a(n)=parsum(k=1, 10^n, isprime(k^2+(k+1)^2));

%Y Cf. A027861, A206709, A006880.

%Y Cf. A199401, A331941.

%K nonn,hard,more

%O 0,2

%A _Hermann Stamm-Wilbrandt_, Jan 04 2026