A206709 - OEIS

A206709

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

%I #56 Feb 24 2020 04:24:09

%S 5,19,112,841,6656,54110,456362,3954181,34900213,312357934,2826683630,

%T 25814570672,237542444180,2199894223892

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

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

%C Examples:

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

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

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

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

%C From _Jacques Tramu_, Sep 14 2018: (Start)

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

%C .

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

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

%C 1 5 3

%C 2 19 17

%C 3 112 115

%C 4 841 843

%C 5 6656 6584

%C 6 54110 53882

%C 7 456362 456175

%C 8 3954181 3954737

%C 9 34900213 34902408

%C 10 312357934 312353959

%C 11 2826683630 2826686358

%C 12 25814570672 25814559712

%C (End)

%C For a comparison with the estimate that results from the Hardy and Littlewood Conjecture F, see A331942. - _Hugo Pfoertner_, Feb 03 2020

%H Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, Stephan Ramon Garcia, <a href="https://arxiv.org/abs/1807.08899">The Bateman-Horn Conjecture: Heuristics, History, and Applications</a>, arXiv:1807.08899 [math.NT], 2018-2019. See Table 2. p. 8.

%H Marek Wolf, <a href="http://arxiv.org/abs/0803.1456">Search for primes of the form m^2+1</a>, arXiv:0803.1456 [math.NT], 2008-2010.

%e 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.

%p 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:

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

%o (PARI) a(n)=sum(n=1,10^n,ispseudoprime(n^2+1)) \\ _Charles R Greathouse IV_, Feb 13 2012

%o (Python)

%o from sympy import isprime

%o def A206709(n):

%o c, b, b2, n10 = 0, 1, 2, 10**n

%o while b <= n10:

%o if isprime(b2):

%o c += 1

%o b += 1

%o b2 += 2*b - 1

%o return c # _Chai Wah Wu_, Sep 17 2018

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

%K nonn,more

%O 1,1

%A _Michel Lagneau_, Feb 13 2012

%E a(11)-a(12) from _Arkadiusz Wesolowski_, Jul 21 2012

%E a(13)-a(14) from _Jinyuan Wang_, Feb 24 2020