A215050
Number of primes of the form 1 + b^16 for 1 < b < 10^n.
Original entry on oeis.org
1, 5, 48, 291, 2194, 17907, 152447, 1322985, 11669082
Offset: 1
a(1) = 1 because the only Fermat prime F_4(b) where b<10^1 is the prime 65537.
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Table[Length[Select[Range[2,10^n-1]^16 + 1, PrimeQ]], {n, 5}] (* T. D. Noe, Aug 02 2012 *)
Module[{nn=8,t},t=Table[If[PrimeQ[n^16+1],1,0],{n,2,10^nn}];Table[Total[ Take[t,10^i-1]],{i,nn}]] (* Harvey P. Dale, Sep 14 2015 *)
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a(n) = sum(b=1, 10^n/2-1, isprime((2*b)^16+1))
A215051
Number of primes of the form 1 + b^32 for 1 < b < 10^n.
Original entry on oeis.org
0, 3, 22, 146, 1062, 8963, 74951, 651537, 5740807, 51389252
Offset: 1
a(2) = 3 because the Fermat numbers F_5(b) where b<10^2 are prime only for b = 30, 54, 96.
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Table[Length[Select[Range[2,10^n-1]^32 + 1, PrimeQ]], {n, 4}] (* T. D. Noe, Aug 01 2012 *)
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a(n) = sum(b=1, 10^n/2-1, isprime((2*b)^32+1))
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
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.
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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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1 + Accumulate@ Array[Count[Range[10^(# - 1) + 1, 10^#], ?(PrimeQ[#^2 + 1] &)] &, 7] (* _Michael De Vlieger, Sep 18 2018 *)
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a(n)=sum(n=1,10^n,ispseudoprime(n^2+1)) \\ Charles R Greathouse IV, Feb 13 2012
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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
A215057
Number of primes of the form 1 + b^64 for 1 < b < 10^n.
Original entry on oeis.org
0, 0, 8, 92, 606, 4835, 41059, 354239, 3133668
Offset: 1
a(3) = 8 because the Fermat numbers F_6(b) where b<10^3 are prime only for b = 102, 162, 274, 300, 412, 562, 592, 728.
A215058
Number of primes of the form 1 + b^128 for 1 < b < 10^n.
Original entry on oeis.org
0, 0, 7, 25, 242, 1933, 16080, 139921, 1234958
Offset: 1
a(3) = 7 because the generalized Fermat numbers F_7(b) where b<10^3 are prime only for b: 120, 190, 234, 506, 532, 548, 960.
A215698
Number of primes of the form 1 + b^256 for 1 < b < 10^n.
Original entry on oeis.org
0, 0, 4, 30, 272, 2322
Offset: 1
a(3) = 4 because the generalized Fermat numbers F_8(b) where b<10^3 are prime only for b: 278, 614, 892, 898.
A215699
Number of primes of the form 1 + b^512 for 1 < b < 10^n.
Original entry on oeis.org
0, 1, 1, 28, 160, 1247
Offset: 1
a(3) = 1 because the generalized Fermat numbers F_9(b) where b<10^3 are prime only for b = 46.
A215700
Number of primes of the form 1 + b^1024 for 1 < b < 10^n.
Original entry on oeis.org
0, 0, 1, 14, 81, 578
Offset: 1
a(3) = 1 because the generalized Fermat numbers F_10(b) where b<10^3 are prime only for b = 824.
A215701
Number of primes of the form 1 + b^2048 for 1 < b < 10^n.
Original entry on oeis.org
0, 0, 1, 4, 40, 276
Offset: 1
a(4) = 4 because the generalized Fermat numbers F_11(b) where b<10^4 are prime only for b = 150, 2558, 4650, 4772.
A215702
Number of primes of the form 1 + b^4096 for 1 < b < 10^n.
Original entry on oeis.org
0, 0, 0, 2, 16, 170
Offset: 1
a(4) = 2 because the generalized Fermat numbers F_12(b) where b<10^4 are prime only for b = 1534, 7316.
Showing 1-10 of 14 results.
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