A259835
a(n) is the number of odd primes of the form b^(2^n)+1 that are less than A123599(n+1).
Original entry on oeis.org
1, 1, 1, 1, 41, 152, 122, 185, 8, 860, 24, 612, 97094
Offset: 0
To find a(5), find all primes b^32 + 1 until you reach a base b that is a perfect square. In this case you find 152 nonsquare b values { 30, 54, 96, 112, ..., 10396 }, but the 153rd b is 10404, a perfect square. So 10404^32 + 1 = 102^64 + 1 belongs to the next family. Therefore a(5)=152.
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b=2;for(n=0,100,x=0;until(,if(ispseudoprime(b^(2^n)+1),if(issquare(b,&b),break,x++));b+=2);print("a(",n,")=",x,", next b is ",b))
A334053
Least b such that b^(2^n) + 1 is a Proth prime (A080076).
Original entry on oeis.org
2, 2, 2, 2, 2, 96, 6912, 960, 16256, 2013184, 235520, 61184, 125440, 992256, 155615232, 550207488, 12192710656
Offset: 0
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a(n) = forstep(b=2,+oo,2,2*valuation(b,2)>logint(b,2)&&ispseudoprime(b^(2^n)+1)&&return(b))
A182065
Smallest average of twin prime pairs s such that s^(2^n)+1 is prime.
Original entry on oeis.org
4, 4, 4, 198, 30, 102, 3000, 7332, 4482, 187218, 150, 114690, 713310, 1943532, 3467622, 4470420, 23045178, 12529818
Offset: 1
198^(2^4)+1 = 5580113648647376991977566450378407937 is prime.
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t=Select[Table[Prime[n]+1,{n,10^5}],PrimeQ[#1+1]&]; s:=t[[m]]; Table[m=1; While[!PrimeQ[s^(2^n)+1],m++]; s,{n,1,9}](* Last five terms obtained by intersection with Yves Gallot records.*)
A277967
Number of even numbers b with 0 < b < 2^n such that b^(2^n) + 1 is prime.
Original entry on oeis.org
0, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 3, 4, 1
Offset: 1
For n=18, we get b^262144 + 1 is prime for b=24518, 40734, 145310, 361658, 525094, ...; the first 3 of these b values are strictly below 262144, hence a(18)=3.
The corresponding primes are 2^4+1; 2^8+1, 4^8+1; 2^16+1; 30^32+1; 120^128+1; 46^512+1; etc.
Cf.
A056993,
A121270,
A259835,
A005574,
A000068,
A006314,
A006313,
A006315,
A006316,
A056994,
A056995,
A057465,
A057002,
A088361,
A088362,
A226528,
A226529,
A226530,
A251597,
A253854,
A244150,
A243959,
A321323, etc.
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Table[Count[Range[2, 2^n - 1, 2], b_ /; PrimeQ[b^(2^n) + 1]], {n, 9}] (* Michael De Vlieger, Nov 10 2016 *)
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a(n)=sum(k=1,2^(n-1)-1,ispseudoprime((2*k)^2^n+1)) \\ slow, only probabilistic primality test
A338931
Least b such that b^(2^n) + 1 is an odd Pierpont prime (A005109).
Original entry on oeis.org
2, 2, 2, 2, 2, 54, 162, 8310407949893763072, 46438023168, 65229815808, 396718580736, 629856, 152461794335880672662217818112
Offset: 0
a(7) corresponds to prime 8310407949893763072^128 + 1 = (2^47*3^10)^128 + 1.
A358621
Smallest b > 1 such that b^(2^n)+1 is a Sophie Germain prime.
Original entry on oeis.org
2, 2, 160, 140, 2800, 8660, 62150, 4085530, 922820, 4629490, 5802710, 1146175000, 90894850
Offset: 0
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a(n)=n<2&&return(2);forstep(b=10,+oo,10,ispseudoprime(b^(2^n)+1)&&ispseudoprime(2*b^(2^n)+3)&&return(b))
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