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A206709 Number of primes of the form b^2 + 1 for b <= 10^n. 11
5, 19, 112, 841, 6656, 54110, 456362, 3954181, 34900213, 312357934, 2826683630, 25814570672, 237542444180, 2199894223892 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..14.

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

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

Sequence in context: A158615 A321652 A088180 * A199480 A209111 A328060

Adjacent sequences:  A206706 A206707 A206708 * A206710 A206711 A206712

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

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Last modified July 10 16:22 EDT 2020. Contains 335577 sequences. (Running on oeis4.)