OFFSET
1,1
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
LINKS
Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, 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.
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Michel Lagneau, Feb 13 2012
EXTENSIONS
a(11)-a(12) from Arkadiusz Wesolowski, Jul 21 2012
a(13)-a(14) from Jinyuan Wang, Feb 24 2020
STATUS
approved