A094493 Primes p such that 2^j+p^j are primes for j=0,1,2,16.

43577, 84317, 93887, 108377, 124247, 346667, 379997, 431867, 461297, 579197, 681257, 819317, 863867, 889037, 1143047, 1146797, 1271027, 1306817, 1518707, 1775867, 1926647, 1948517, 2119937, 2177447, 2348807, 2491607, 2604557

Offset: 1

Author

Labos Elemer, Jun 01 2004

Keywords

• nonn

Comments

Primes of 2^j+p^j form are a generalization of Fermat-primes. 1^j is replaced by p^j. This is strongly supported by the observation that corresponding j-exponents are apparently powers of 2 like for the 5 known Fermat primes. See A094473-A094491.

Examples

			For j=0: 1+1=2 is prime; other conditions are:
because of p^1+2=prime; 3rd and 4th conditions are as
follows: prime=p^2+4 and prime=65536+p^16.
		

Crossrefs

Cf. A082101, A094473-A094491.

Programs

Mathematica

{ta=Table[0, {100}], u=1}; Do[s0=2;s1=2+Prime[j]^1;s2=4+Prime[j]^2;s16=65536+Prime[j]^16 If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s16], Print[{j, Prime[j]}];ta[[u]]=Prime[j];u=u+1], {j, 1, 1000000}]
Select[Prime[Range[2*10^5]],AllTrue[Table[2^k+#^k,{k,{0,1,2,16}}],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 05 2021 *)