A206709 -id:A206709 - cp's OEIS frontend

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

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

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.

A302434 Number of primes of the form b^2 + 2 for b <= 10^n.

Original entry on oeis.org

4, 12, 69, 447, 3423, 27869, 236985, 2054022, 18127693, 162237123

Offset: 1

Seiichi Manyama, Apr 07 2018

From Jacques Tramu, Sep 13 2018: (Start)
Table C(i) = a(i)/(n*log(n)), with n = 10^i:
a(1)  =         4     C(1)  = 0.92103404
a(2)  =        12     C(2)  = 0.55262042
a(3)  =        69     C(3)  = 0.47663511
a(4)  =       447     C(4)  = 0.41170221
a(5)  =      3423     C(5)  = 0.39408744
a(6)  =     27869     C(6)  = 0.38502446
a(7)  =    236985     C(7)  = 0.38197469
a(8)  =   2054022     C(8)  = 0.37836484
a(9)  =  18127693     C(9)  = 0.37566500
a(10) = 162237123     C(10) = 0.37356478
(End)

			a(1) = 4 because there are 4 primes of the form b^2+2 for b <= 10: 2, 3, 11 and 83.

Number of primes of the form b^2+m for b <= 10^n: A302443 (m=-3), A302442 (m=-2), A206709 (m=1), this sequence (m=2), A302435 (m=3).

Cf. A056899.

PARI
```
{a(n) = sum(k=0, 10^n, isprime(k^2+2))}
```

a(10) from Jacques Tramu, Sep 13 2018

A302435 Number of primes of the form b^2+3 for b <= 10^n.

Original entry on oeis.org

5, 18, 110, 712, 5427, 44096, 373019, 3228862, 28494961

Offset: 1

Seiichi Manyama, Apr 07 2018

			a(1) = 5 because there are 5 primes of the form b^2+3 for b <= 10: 3, 7, 19, 67 and 103.

Number of primes of the form b^2+m for b <= 10^n: A302443 (m=-3), A302442 (m=-2), A206709 (m=1), A302434 (m=2), this sequence (m=3).

Cf. A049423.

PARI
```
{a(n) = sum(k=0, 10^n, isprime(k^2+3))}
```

A302442 Number of primes of the form b^2-2 for b <= 10^n.

Original entry on oeis.org

5, 26, 157, 1153, 8888, 72928, 615643, 5328644, 47034083, 420950239

Offset: 1

Seiichi Manyama, Apr 08 2018

From Jacques Tramu, Sep 13 2018: (Start)
Table C(i) = a(i)/pi(10^i) = a(i)/A000720(10^i)
a(1)  =         5         C(1)  = 1.25000000
a(2)  =        26         C(2)  = 1.04000000
a(3)  =       157         C(3)  = 0.93452381
a(4)  =      1153         C(4)  = 0.93816111
a(5)  =      8888         C(5)  = 0.92660550
a(6)  =     72928         C(6)  = 0.92904278
a(7)  =    615643         C(7)  = 0.92636541
a(8)  =   5328644         C(8)  = 0.92487818
a(9)  =  47034083         C(9)  = 0.92500224
a(10) = 420950239         C(10) = 0.92505860
(End)

			a(1) = 5 because there are 5 primes of the form b^2-2 for b <= 10 : 2, 7, 23, 47 and 79.

Number of primes of the form b^2+m for b <= 10^n: A302443 (m=-3), this sequence (m=-2), A206709 (m=1), A302434 (m=2), A302435 (m=3).

Cf. A028871.

{a(n) = sum(k=0, 10^n, isprime(k^2-2))}

from sympy import isprime
def aupton(terms):
  s, alst = 0, []
  for n in range(1, terms+1):
    s += sum(isprime(b**2-2) for b in range(10**(n-1), 10**n))
    alst.append(s)
  return alst
print(aupton(6)) # Michael S. Branicky, May 26 2021

a(10) from Jacques Tramu, Sep 14 2018

A302443 Number of primes of the form b^2-3 for b <= 10^n.

Original entry on oeis.org

3, 19, 119, 849, 6663, 54514, 460019, 3982973, 35174007

Offset: 1

Seiichi Manyama, Apr 08 2018

			a(1) = 3 because there are 3 primes of the form b^2-3 for b <= 10 : 13, 61 and 97.

Number of primes of the form b^2+m for b <= 10^n: this sequence (m=-3), A302442 (m=-2), A206709 (m=1), A302434 (m=2), A302435 (m=3).

Cf. A028874.

PARI
```
{a(n) = sum(k=0, 10^n, isprime(k^2-3))}
```

A301943 Number of primes of the form b^2+1 for b <= 10^n that end in 1.

Original entry on oeis.org

1, 4, 42, 279, 2236, 18155, 152020, 1317648, 11634451, 104116591, 942191087

Offset: 1

Seiichi Manyama, Mar 29 2018

			101, 401, 1601 and 8101 are primes; so a(2) = 4.

Cf. A002496, A206709, A301944.

c = k = 0; lst = {}; Do[ While[k <= 10^n, If[ PrimeQ[k^2 + 1], c++]; k+=10]; AppendTo[lst, c]; Print[c], {n, 9}] (* Robert G. Wilson v, Mar 30 2018 *)

from sympy import isprime
def A301943(n):
    return sum(1 for i in range(1,10**(n-1)+1) if isprime(100*i**2+1)) # Chai Wah Wu, Mar 30 2018

a(n) + A301944(n) + 2 = A206709(n).

a(10) from Robert G. Wilson v, Mar 31 2018

a(11) from Robert G. Wilson v, Apr 04 2018

A301944 Number of primes of the form b^2 + 1 for b <= 10^n that end in 7.

Original entry on oeis.org

2, 13, 68, 560, 4418, 35953, 304340, 2636531, 23265760, 208241341, 1884492541

Offset: 1

Seiichi Manyama, Mar 29 2018

			17, 37, 197, 257, 577, 677, 1297, 2917, 3137, 4357, 5477, 7057 and 8837 are primes; so a(2) = 13.

Cf. A002496, A206709, A301943.

a(n) + A301943(n) + 2 = A206709(n).

a(10)-a(11) from Jinyuan Wang, Feb 24 2020

A319228 Number of primes of the form b^2 + b + 1 for b <= 10^n.

Original entry on oeis.org

6, 32, 189, 1410, 10751, 88118, 745582, 6456835, 56988601, 510007598, 4615215645

Offset: 1

Seiichi Manyama, Sep 14 2018

			a(1) = 6 because there are 6 primes of the form b^2 + b + 1 for b <= 10: 3, 7, 13, 31, 43 and 73.

Cf. A002383, A002384, A206709.

{a(n) = sum(k=0, 10^n, isprime(k^2+k+1))}

from sympy import isprime
def A319228(n):
    c, b, b2, n10 = 0, 1, 3, 10**n
    while b <= n10:
        if isprime(b2):
            c += 1
        b += 1
        b2 += 2*b
    return c # Chai Wah Wu, Sep 17 2018

a(10) from Chai Wah Wu, Sep 17 2018

a(11) from Chai Wah Wu, Sep 18 2018

A300363 Sum of primes of the form b^2+1 for b <= 10^n.

Original entry on oeis.org

2, 162, 45052, 29570385, 24699073806, 20281853994629, 16998638126185703, 14511042624337529267, 12647180923917812683382, 11222444317042682292853518, 10085250548770665025465417675

Offset: 0

Seiichi Manyama, Mar 04 2018

			a(1) = 2 + 5 + 17 + 37 + 101 = 162.
a(2) = 2 + 5 + 17 + 37 + 101 + 197 + 257 + 401 + 577 + 677 + 1297 + 1601 + 2917 + 3137 + 4357 + 5477 + 7057 + 8101 + 8837 = 45052.

Cf. A002496, A206709.

list(len) = {my(pow = 1, p, s = 0); for(k = 1, 10^len, p = k^2 + 1; if(isprime(p), s += p); if(k == pow, print1(s, ", "); pow *= 10));} \\ Amiram Eldar, Jul 18 2025

a(8) corrected and a(9)-a(10) added by Amiram Eldar, Jul 18 2025

cp's OEIS Frontend

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

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

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A302434 Number of primes of the form b^2 + 2 for b <= 10^n.

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A302435 Number of primes of the form b^2+3 for b <= 10^n.

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A302442 Number of primes of the form b^2-2 for b <= 10^n.

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A302443 Number of primes of the form b^2-3 for b <= 10^n.

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A301943 Number of primes of the form b^2+1 for b <= 10^n that end in 1.

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A301944 Number of primes of the form b^2 + 1 for b <= 10^n that end in 7.

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