A392244 Number of primes of the form b^2 + (b+1)^2 for b <= 10^n.

1, 6, 36, 225, 1645, 12706, 104894, 892723, 7755330, 68588950, 614983774, 5573589175

Offset

0,2

Comments

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

== ===============

n a(n)/A206709(n)

== ===============

0 1

1 1.2

2 1.894737

3 2.008929

4 1.956005

5 1.908954

6 1.938537

7 1.956173

8 1.961299

9 1.965287

10 1.968843

11 1.97178

.

Conjecture 2 (conjecture 1 + A206709 conjecture):

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

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

Links

Table of n, a(n) for n=0..11.

Formula

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

Example

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.

Prog

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

Crossrefs

Cf. A027861, A206709, A006880.

Cf. A199401, A331941.

Sequence in context: A166748 A200378 A085687 * A242136 A129327 A244889

Adjacent sequences: A392241 A392242 A392243 * A392245 A392246 A392247

Keyword

nonn,hard,more

Author

Hermann Stamm-Wilbrandt, Jan 04 2026

Status

approved