A343120 a(n) is the smallest number b such that (b^(2^k) + 1)/2 is prime for k = 0, 1, ..., n.
3, 3, 3, 205, 2326161, 20589460461, 3847314721101
Offset: 0
Examples
For n=3, the four numbers (205+1)/2, (205^2+1)/2, (205^4+1)/2, and (205^8+1)/2 are prime, and 205 is smallest with this property, so a(3)=205.
Programs
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PARI
a(n)=forstep(b=3,+oo,2,for(k=0,n,!ispseudoprime((b^(2^k)+1)/2)&&next(2));return(b)) \\ if a(n-1) is known, b loop can start from there instead
Extensions
a(6) found by Kellen Shenton added by Jeppe Stig Nielsen, Apr 09 2021