A083844 -id:A083844 - cp's OEIS frontend

A331942 a(n) = number of primes of the form P(k) = k^2 + 1 <= 10^n as predicted by the Hardy and Littlewood Conjecture F, rounded to nearest integer. The actual number of primes is A083844(n).

1, 4, 9, 20, 48, 121, 317, 855, 2356, 6609, 18787, 53970, 156385, 456404, 1340088, 3955219, 11726332, 34903256, 104251560, 312353236, 938461459, 2826668497, 8533343468, 25814350227, 78239112814, 237541788793, 722354115787, 2199893807666, 6708847354653, 20485514756657

Offset: 1

Hugo Pfoertner, Feb 02 2020

nonn

Comparison of actual and approximated number of primes < 10^n:
   Limit
   10^n
    | A083844(n)
    |   | a(n)
    |   | |   (a(n) - A083844(n))/A083844(n)
   10^1 2 1  -0.50000
   10^2 4 4   0.0
   10^3 10 9  -0.10000
   10^4 19 20  0.052632
   10^5 51 48  -0.058824
   10^6 112 121  0.080357
   10^7 316 317  0.0031646
   10^8 841 855  0.016647
   10^9 2378 2356  -0.0092515
  10^10 6656 6609  -0.0070613
  10^11 18822 18787  -0.0018595
  10^12 54110 53970  -0.0025873
  10^13 156081 156385  0.0019477
  10^14 456362 456404  9.2032E-5
  10^15 1339875 1340088  0.00015897
  10^16 3954181 3955219  0.00026251
  10^17 11726896 11726332  -4.8095E-5
  10^18 34900213 34903256   8.7191E-5
  10^19 104248948 104251560   2.5055E-5
  10^20 312357934 312353236  -1.5040E-5
  10^21 938457801 938461459   3.8979E-6
  10^22 2826683630 2826668497  -5.3536E-6
  10^23 8533327397 8533343468   1.8833E-6
  10^24 25814570672 25814350227  -8.5396E-6
  10^25 78239402726 78239112814  -3.7054E-6
  10^26 237542444180 237541788793  -2.7590E-6
  10^27 722354138859 722354115787  -3.1940E-8
  10^28 2199894223892 2199893807666  -1.8920E-7

Keith Conrad, Hardy-Littlewood Constants, (2003).

Cf. A083844, A331941.

C=0.68640673140912300455609634836350943408916655062787977896811707366392;
x=1.0;S10=sqrt(10);for(k=1,30,x*=s10;print1(round(C*intnum(y=2,x,1/log(y))),", "))

b(m) = round (C * Integral_{x=2..m} 1/log(x) dx), with C ~= 0.6864067314..., the Hardy-Littlewood constant for k^2 + 1 (A331941); a(n) = b(10^(n/2)).

A002496 Primes of the form k^2 + 1.

Original entry on oeis.org

2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177

Offset: 1

N. J. A. Sloane

It is conjectured that this sequence is infinite, but this has never been proved.
An equivalent description: primes of form P = (p1*p2*...*pm)^k + 1 where p1..pm are primes and k > 1, since then k must be even for P to be prime.
Also prime = p(n) if A054269(n) = 1, i.e., quotient-cycle-length = 1 in continued fraction expansion of sqrt(p). - Labos Elemer, Feb 21 2001
Also primes p such that phi(p) is a square.
Also primes of form x*y + z, where x, y and z are three successive numbers. - Giovanni Teofilatto, Jun 05 2004
It is a result that goes back to Mirsky that the set of primes p for which p-1 is squarefree has density A, where A = A005596 denotes the Artin constant. More precisely, Sum_{p <= x} mu(p-1)^2 = A*x/log x + o(x/log x) as x tends to infinity. Conjecture: Sum_{p <= x, mu(p-1)=1} 1 = (A/2)*x/log x + o(x/log x) and Sum_{p <= x, mu(p-1)=-1} 1 = (A/2)*x/log x + o(x/log x). - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003
Also primes of the form x^y + 1, where x > 0, y > 1. Primes of the form x^y - 1 (x > 0, y > 1) are the Mersenne primes listed in A000668(n) = {3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...}. - Alexander Adamchuk, Mar 04 2007
With the exception of the first two terms {2,5}, the continued fraction (1 + sqrt(p))/2 has period 3. - Artur Jasinski, Feb 03 2010
With the exception of the first term {2}, congruent to 1 (mod 4). - Artur Jasinski, Mar 22 2011
With the exception of the first two terms, congruent to 1 or 17 (mod 20). - Robert Israel, Oct 14 2014
From Bernard Schott, Mar 22 2019: (Start)
These primes are the primitive terms which generate the sequence of integers with only one prime factor and whose Euler's totient is a square: A054755. So this sequence is a subsequence of A054755 and of A039770. Additionally, the terms of this sequence also have a square cototient, so this sequence is a subsequence of A063752 and A054754.
If p prime = n^2 + 1, phi(p) = n^2 and cototient(p) = 1^2.
Except for 3, the four Fermat primes in A019434 {5, 17, 257, 65537}, belong to this sequence; with F_k = 2^(2^k) + 1, phi(F_k) = (2^(2^(k-1)))^2.
See the file "Subfamilies and subsequences" (& I) in A039770 for more details, proofs with data, comments, formulas and examples. (End)
In this sequence, primes ending with 7 seem to appear twice as often as primes ending with 1. This is because those with 7 come from integers ending with 4 or 6, while those with 1 come only from integers ending with 0 (see De Koninck & Mercier reference). - Bernard Schott, Nov 29 2020
The set of odd primes p for which every elliptic curve of the form y^2 = x^3 + d*x has order p-1 over GF(p) for those d with (d,p)=1 and d a fourth power modulo p. - Gary Walsh, Sep 01 2021 [edited, Gary Walsh, Apr 26 2025]

Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 211 pp. 34 and 169, Ellipses, Paris, 2004.
Leonhard Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 22.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
Hugh L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
C. Stanley Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 116.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 134.

T. D. Noe, Table of n, a(n) for n = 1..10000
Tewodros Amdeberhan, Luis A. Medina and Victor H. Moll, Arithmetical properties of a sequence arising from an arctangent sum, J. Numb. Theory, Vol. 128, No. 6 (2008), pp. 1807-1846, eq. (1.10).
William D. Banks, John B. Friedlander, Carl Pomerance and Igor E. Shparlinski, Multiplicative structure of values of the Euler function, in High Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams (A. Van der Poorten, ed.), Fields Inst. Comm. 41 (2004), pp. 29-47.
Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation, Vol. 16, No. 79 (1962), pp. 363-367.
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See p. 11.
Frank Ellermann, Primes of the form (m^2)+1 up to 10^6.
Dan Ismailescu and Yunkyu James Lee, Polynomially growing integer sequences all whose terms are composite, arXiv:2501.04851 [math.NT], 2025. See p. 9.
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, Amer. Math. Monthly, Vol. 56, No. 1 (1949), pp. 17-19.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Apoloniusz Tyszka and Sławomir Kurpaska, Open problems that concern computable sets X, subset of N, and cannot be formally stated as they refer to current knowledge about X, (2020).
Eric Weisstein's World of Mathematics, Landau's Problems.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Wikipedia, Bateman-Horn Conjecture.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A083844 (number of these primes < 10^n), A199401 (growth constant).
Cf. A001912, A005574, A054964, A062325, A088179, A090693, A141293, A172168.
Cf. A000668 (Mersenne primes), A019434 (Fermat primes).
Subsequence of A039770.
Cf. A010051, subsequence of A002522.
Cf. A237040 (an analog for n^3 + 1).
Cf. A010051, A000290; subsequence of A028916.
Subsequence of A039770, A054754, A054755, A063752.
Primes of form n^2+b^4, b fixed: A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).
Cf. A030430 (primes ending with 1), A030432 (primes ending with 7).

a002496 n = a002496_list !! (n-1)
a002496_list = filter ((== 1) . a010051') a002522_list
-- Reinhard Zumkeller, May 06 2013

[p: p in PrimesUpTo(100000)| IsSquare(p-1)]; // Vincenzo Librandi, Apr 09 2011

select(isprime, [2, seq(4*i^2+1, i= 1..1000)]); # Robert Israel, Oct 14 2014

Select[Range[100]^2+1, PrimeQ]
Join[{2},Select[Range[2,300,2]^2+1,PrimeQ]] (* Harvey P. Dale, Dec 18 2018 *)

isA002496(n) = isprime(n) && issquare(n-1) \\ Michael B. Porter, Mar 21 2010

is_A002496(n)=issquare(n-1)&&isprime(n) \\ For "random" numbers in the range 10^10 and beyond, at least 5 times faster than the above. - M. F. Hasler, Oct 14 2014

# Python 3.2 or higher required
from itertools import accumulate
from sympy import isprime
A002496_list = [n+1 for n in accumulate(range(10**5),lambda x,y:x+2*y-1) if isprime(n+1)] # Chai Wah Wu, Sep 23 2014

# Python 2.4 or higher required
from sympy import isprime
A002496_list = list(filter(isprime, (n*n+1 for n in range(10**5)))) # David Radcliffe, Jun 26 2016

There are O(sqrt(n)/log(n)) terms of this sequence up to n. But this is just an upper bound. See the Bateman-Horn or Wolf papers, for example, for the conjectured for what is believed to be the correct density.
a(n) = 1 + A005574(n)^2. - R. J. Mathar, Jul 31 2015
Sum_{n>=1} 1/a(n) = A172168. - Amiram Eldar, Nov 14 2020
a(n+1) = 4*A001912(n)^2 + 1. - Hal M. Switkay, Apr 03 2022

Formula, reference, and comment from Charles R Greathouse IV, Aug 24 2009

Edited by M. F. Hasler, Oct 14 2014

A215047 Number of primes of the form 1 + b^2 for 1 < b < 10^n.

Original entry on oeis.org

3, 18, 111, 840, 6655, 54109, 456361, 3954180, 34900212, 312357933, 2826683629, 25814570671, 237542444179, 2199894223891

Offset: 1

Henryk Dabrowski, Aug 01 2012

Primes 1 + b^2 are a form of generalized Fermat primes.

It is conjectured that a(n) is asymptotic to 0.6864067*li(10^n).

			a(1) = 3 because the only generalized Fermat primes F_1(b) where b < 10^1 are the primes: 5, 17, 37.

Yves Gallot, Status of the smallest base values yielding Generalized Fermat primes
Yves Gallot, How many prime numbers appear in a sequence ?
Yves Gallot, A Problem on the Conjecture Concerning the Distribution of Generalized Fermat Prime numbers (a new method for the search for large primes)
Mersenne Wiki, Table of known GF primes b^n+1 where n (exponent) is at least 8192.
Daniel Shanks, On the Conjecture of Hardy & Littlewood concerning the Number of Primes of the Form n^2 + a, Math. Comp. 14 (1960), 320-332.

Cf. A083844, A206709.

Table[Length[Select[Range[2,10^n-1]^2 + 1, PrimeQ]], {n, 5}] (* T. D. Noe, Aug 02 2012 *)

a(n) = sum(b=1,10^n/2-1,isprime((2*b)^2+1))

a(n) = A083844(2*n) - 1.

a(13)-a(14) from Jinyuan Wang, Feb 23 2020

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

Original entry on oeis.org

5, 19, 112, 841, 6656, 54110, 456362, 3954181, 34900213, 312357934, 2826683630, 25814570672, 237542444180, 2199894223892

Offset: 1

Michel Lagneau, Feb 13 2012

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

			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.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 264.

Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and 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.

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

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:

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

a(n)=sum(n=1,10^n,ispseudoprime(n^2+1)) \\ Charles R Greathouse IV, Feb 13 2012

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

a(11)-a(12) from Arkadiusz Wesolowski, Jul 21 2012

a(13)-a(14) from Jinyuan Wang, Feb 24 2020

A331941 Hardy-Littlewood constant for the polynomial x^2 + 1.

Original entry on oeis.org

6, 8, 6, 4, 0, 6, 7, 3, 1, 4, 0, 9, 1, 2, 3, 0, 0, 4, 5, 5, 6, 0, 9, 6, 3, 4, 8, 3, 6, 3, 5, 0, 9, 4, 3, 4, 0, 8, 9, 1, 6, 6, 5, 5, 0, 6, 2, 7, 8, 7, 9, 7, 7, 8, 9, 6, 8, 1, 1, 7, 0, 7, 3, 6, 6, 3, 9, 2, 1, 1, 1, 3, 3, 5, 8, 6, 8, 5, 1, 1, 5, 8, 6, 3, 8, 5, 9

Offset: 0

Hugo Pfoertner, Feb 02 2020

			0.686406731409123004556096348363509434089166550627879778968117...

Henri Cohen, Number Theory, Vol II: Analytic and Modern Tools, Springer (Graduate Texts in Mathematics 240), 2007.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 85.

Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998). [pdf copy, with permission]
Keith Conrad, Hardy-Littlewood Constants, (2003).

Cf. A002496, A005574, A083844, A199401, A206709, A221712.

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+1)/2 after setting the required precision.

Equals (1/2)*Product_{p=primes} (1 - Kronecker(-4,p)/(p - 1)).

Equals A199401/2.

A083849 a(n) is the largest prime of the form x^2 + 1 <= 2^n.

Original entry on oeis.org

2, 2, 5, 5, 17, 37, 101, 197, 401, 677, 1601, 3137, 8101, 15877, 32401, 62501, 122501, 246017, 512657, 1020101, 2073601, 4137157, 8386817, 16695397, 33339077, 66977857, 133772357, 268304401, 536663557, 1073610757, 2146098277

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that this sequence is increasing, but this has never been proved.

It is easily shown that all terms greater than 5 end in 1 or 7.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A005574, A002496, A083844, A083845, A083846, A083847, A083848.

a(n) = my(last = 2^n+1); while ((p = precprime(last-1)) && (! issquare(p-1)), last = p;); p \\ Michel Marcus, Jun 14 2013

a(n)=my(k=sqrtint(2^n-1)); while(!isprime(k^2+1), k--); k^2+1 \\ Charles R Greathouse IV, Nov 29 2013

A083847 a(n) = number of primes of the form x^2 + 1 <= 2^n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 24, 33, 42, 54, 70, 91, 114, 158, 212, 293, 393, 539, 713, 957, 1301, 1792, 2459, 3378, 4615, 6233, 8418, 11540, 15867, 21729, 29843, 41169, 56534, 77697, 106787, 147067, 203025, 280340, 387308, 535153, 739671, 1023655, 1416635, 1960813, 2716922, 3764693, 5218926, 7238715

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that the number of primes of the form x^2+1 is infinite and thus this sequence does not become a constant, but this has never been proved.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

C. K. Caldwell, An amazing prime heuristic.
Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A005574, A002496, A083844 - A083849.

a(n) = my(nb = 0); forprime(p=2, 2^n, if (issquare(p-1), nb++);); nb  \\ Michel Marcus, Jun 14 2013

More terms from Alexander D. Healy, Feb 06 2005

A199401 Decimal expansion of constant Product_{p>=3} (1 - (-1)^((p-1)/2)/(p-1)). Hardy-Littlewood constant of x^2 + 1.

Original entry on oeis.org

1, 3, 7, 2, 8, 1, 3, 4, 6, 2, 8, 1, 8, 2, 4, 6, 0, 0, 9, 1, 1, 2, 1, 9, 2, 6, 9, 6, 7, 2, 7, 0, 1, 8, 8, 6, 8, 1, 7, 8, 3, 3, 3, 1, 0, 1, 2, 5, 5, 7, 5, 9, 5, 5, 7, 9, 3, 6, 2, 3, 4, 1, 4, 7, 3, 2, 7, 8, 4, 2, 2, 2, 6, 7, 1, 7, 3, 7, 0, 2, 3, 1, 7, 2, 7, 7, 1

Offset: 1

N. J. A. Sloane, Nov 05 2011

Arises in studying A002496.

The constant is Product_{primes p} (1-chi(p)/(p-1)) where chi is the Dirichlet character A101455. Its Euler expansion is (1/(L(m=4,r=2,s=1)* zeta(m=4,n=3,s=2)) *Product_{s>=2} zeta(m=4,n=1,s)^gamma(s), where L and zeta are the functions tabulated in arXiv:1008.2547 and gamma is the sequence A001037. In particular L(m=4,r=2,s=1) = A003881 and zeta(m=4,n=1,s=2)=A175647. - R. J. Mathar, Nov 29 2011

			1.372813462818246009112192696727...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 85.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 264.

T. Amdeberhan, L. A. Median, and V. H. Moll, Arithmetical properties of a sequence arising from an arctangent sum, J. Numb. Theory 128 (2008) 1807-1846, eq. (1.10).
Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998). [pdf copy, with permission]
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70. See Section 5.41.
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A002496.
Cf. A001037, A003881, A083844, A175647, A221712, A331942.
Equals 2*constant given by A331941.

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+1) after setting the required precision.

Extended title, a(30) and beyond from Hugo Pfoertner, Feb 16 2020

cp's OEIS Frontend

A331942 a(n) = number of primes of the form P(k) = k^2 + 1 <= 10^n as predicted by the Hardy and Littlewood Conjecture F, rounded to nearest integer. The actual number of primes is A083844(n).

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A002496 Primes of the form k^2 + 1.

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A215047 Number of primes of the form 1 + b^2 for 1 < b < 10^n.

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

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A331941 Hardy-Littlewood constant for the polynomial x^2 + 1.

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A083849 a(n) is the largest prime of the form x^2 + 1 <= 2^n.

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