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A206709
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Number of primes of the form b^2 + 1 for b <= 10^n.
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11
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5, 19, 112, 841, 6656, 54110, 456362, 3954181, 34900213, 312357934, 2826683630, 25814570672, 237542444180, 2199894223892
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OFFSET
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1,1
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COMMENTS
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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...
In the table below, K = 0.686413 and pi(10^n) = A000720(10^n):
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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
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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1 + Accumulate@ Array[Count[Range[10^(# - 1) + 1, 10^#], _?(PrimeQ[#^2 + 1] &)] &, 7] (* Michael De Vlieger, Sep 18 2018 *)
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PROG
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(Python)
from sympy import isprime
c, b, b2, n10 = 0, 1, 2, 10**n
while b <= n10:
if isprime(b2):
c += 1
b += 1
b2 += 2*b - 1
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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