cp's OEIS Frontend

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

%I A206709 #62 Apr 25 2025 16:00:07
%S A206709 5,19,112,841,6656,54110,456362,3954181,34900213,312357934,2826683630,
%T A206709 25814570672,237542444180,2199894223892
%N A206709 Number of primes of the form b^2 + 1 for b <= 10^n.
%C A206709 Conjecture: The number of primes of the form b^2 + 1 and less than n is asymptotic to 3*n/(4*log(n)).
%C A206709 Examples:
%C A206709 n = 10^3, a(n) = 112 and 3*10^3/(4*log(10^3)) = 108.573...;
%C A206709 n = 10^4, a(n) = 841 and 3*10^4/(4*log(10^4)) = 814.302...;
%C A206709 n = 10^10, a(n) = 312357934 and 3*10^10/(4*log(10^10)) = 325720861.42...
%C A206709 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 A206709 From _Jacques Tramu_, Sep 14 2018: (Start)
%C A206709 In the table below, K = 0.686413 and pi(10^n) = A000720(10^n):
%C A206709 .
%C A206709    n         a(n)   K*pi(10^n)
%C A206709   ==  ===========  ===========
%C A206709    1            5            3
%C A206709    2           19           17
%C A206709    3          112          115
%C A206709    4          841          843
%C A206709    5         6656         6584
%C A206709    6        54110        53882
%C A206709    7       456362       456175
%C A206709    8      3954181      3954737
%C A206709    9     34900213     34902408
%C A206709   10    312357934    312353959
%C A206709   11   2826683630   2826686358
%C A206709   12  25814570672  25814559712
%C A206709 (End)
%C A206709 For a comparison with the estimate that results from the Hardy and Littlewood Conjecture F, see A331942. - _Hugo Pfoertner_, Feb 03 2020
%D A206709 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 264.
%H A206709 Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and 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 A206709 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 A206709 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 A206709 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 A206709 1 + Accumulate@ Array[Count[Range[10^(# - 1) + 1, 10^#], _?(PrimeQ[#^2 + 1] &)] &, 7] (* _Michael De Vlieger_, Sep 18 2018 *)
%o A206709 (PARI) a(n)=sum(n=1,10^n,ispseudoprime(n^2+1)) \\ _Charles R Greathouse IV_, Feb 13 2012
%o A206709 (Python)
%o A206709 from sympy import isprime
%o A206709 def A206709(n):
%o A206709     c, b, b2, n10 = 0, 1, 2, 10**n
%o A206709     while b <= n10:
%o A206709         if isprime(b2):
%o A206709             c += 1
%o A206709         b += 1
%o A206709         b2 += 2*b - 1
%o A206709     return c # _Chai Wah Wu_, Sep 17 2018
%Y A206709 Cf. A002496, A083844, A215047, A331941, A331942.
%K A206709 nonn,more
%O A206709 1,1
%A A206709 _Michel Lagneau_, Feb 13 2012
%E A206709 a(11)-a(12) from _Arkadiusz Wesolowski_, Jul 21 2012
%E A206709 a(13)-a(14) from _Jinyuan Wang_, Feb 24 2020