A206709 Number of primes of the form b^2 + 1 for b <= 10^n.

5, 19, 112, 841, 6656, 54110, 456362, 3954181, 34900213, 312357934, 2826683630, 25814570672, 237542444180, 2199894223892

Offset: 1

Author

Michel Lagneau, Feb 13 2012

Keywords

• nonn
• more

Comments

Conjecture: The number of primes of the form b^2 + 1 and less than n is asymptotic to 3*n/(4*log(n)).

Examples:

n = 10^3, a(n) = 112 and 3*10^3/(4*log(10^3)) = 108.573...;

n = 10^4, a(n) = 841 and 3*10^4/(4*log(10^4)) = 814.302...;

n = 10^10, a(n) = 312357934 and 3*10^10/(4*log(10^10)) = 325720861.42...

a(n) = A083844(2*n), but not always! The only known exception to this rule is at n = 1. - Arkadiusz Wesolowski, Jul 21 2012

From Jacques Tramu, Sep 14 2018: (Start)

In the table below, K = 0.686413 and pi(10^n) = A000720(10^n):

.

n a(n) K*pi(10^n)

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

1 5 3

2 19 17

3 112 115

4 841 843

5 6656 6584

6 54110 53882

7 456362 456175

8 3954181 3954737

9 34900213 34902408

10 312357934 312353959

11 2826683630 2826686358

12 25814570672 25814559712

(End)

For a comparison with the estimate that results from the Hardy and Littlewood Conjecture F, see A331942. - Hugo Pfoertner, Feb 03 2020

Examples

			a(2) = 19 because there are 19 primes of the form b^2 + 1 for b less than 10^2: 2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101 and 8837.
		

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 264.

Links

• Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Table 2. p. 8.
• Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Crossrefs

Cf. A002496, A083844, A215047, A331941, A331942.

Programs

Maple

for n from 1 to 9 do : i:=0:for m from 1 to 10^n do:x:=m^2+1:if type(x,prime)=true then i:=i+1:else fi:od: printf ( "%d %d \n",n,i):od:

Mathematica

1 + Accumulate@ Array[Count[Range[10^(# - 1) + 1, 10^#], ?(PrimeQ[#^2 + 1] &)] &, 7] (* _Michael De Vlieger, Sep 18 2018 *)

PARI

a(n)=sum(n=1,10^n,ispseudoprime(n^2+1)) \\ Charles R Greathouse IV, Feb 13 2012

Python

from sympy import isprime
def A206709(n):
    c, b, b2, n10 = 0, 1, 2, 10**n
    while b <= n10:
        if isprime(b2):
            c += 1
        b += 1
        b2 += 2*b - 1
    return c # Chai Wah Wu, Sep 17 2018

Extensions

a(11)-a(12) from Arkadiusz Wesolowski, Jul 21 2012

a(13)-a(14) from Jinyuan Wang, Feb 24 2020