A176680 -id:A176680 - cp's OEIS frontend

A269018 Primes p of the form 2^k + 2*(-1)^k - 1.

2, 5, 17, 29, 257, 509, 65537, 536870909, 13164036458569648337239753460458804039861886925068638906788872189, 3369993333393829974333376885877453834204643052817571560137951281149, 13803492693581127574869511724554050904902217944340773110325048447598589

Offset: 1

Jaroslav Krizek, Feb 17 2016

nonn

Corresponding values of k: 0, (2, 3), 4, 5, 8, 9, 16, 29, 213, 221, 233, ...; for the prime 5 there are two values: 2 and 3.
Fermat primes > 3 from A019434 are terms.
Prime terms from A269019.
Conjecture: union of {2}, {A019434(n) for n > 1} and {A176680(n)}.
a(16) > 2^16000 if it exists. - Robert Israel, Nov 11 2022
a(16) = 2^20757 - 3, a(17) = 2^30041 - 3. a(18) > 2^40000, if it exists. - Jon E. Schoenfield, Nov 11 2022

Robert Israel, Table of n, a(n) for n = 1..15

Cf. A019434, A052539, A092506, A176680, A267921, A269019.

Set(Sort([2^n + 2*(-1)^n - 1: n in [0..300] | IsPrime(2^n + 2*(-1)^n - 1)]))

Res:= 2,5: count:= 2:
for n from 4 while count < 15 do
   x:= 2^n + 2*(-1)^n - 1;
   if isprime(x) then Res:= Res,x; count:= count+1  fi;
od:
Res; # Robert Israel, Nov 11 2022

A269019 a(n) = 2^n + 2*(-1)^n - 1.

Original entry on oeis.org

2, -1, 5, 5, 17, 29, 65, 125, 257, 509, 1025, 2045, 4097, 8189, 16385, 32765, 65537, 131069, 262145, 524285, 1048577, 2097149, 4194305, 8388605, 16777217, 33554429, 67108865, 134217725, 268435457, 536870909, 1073741825, 2147483645, 4294967297, 8589934589

Offset: 0

Jaroslav Krizek, Feb 17 2016

sign

Fermat numbers > 3 from A000215 are terms.
Prime terms are in A269018.
Union of A052539 and A267921.

			For n = 6; a(n) = 2^n + 2*(-1)^n - 1 = 2^6 + 2*(-1)^6 - 1 = 65.

Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A019434, A052539, A092506, A176680, A267921, A269018.

Magma
```
[2^n + 2*(-1)^n - 1: n in [0..300]]
```

Table [2^n + 2 (-1)^n - 1, {n, 0, 80}] (* or *) CoefficientList[Series[(2 - 5 x + 5 x^2) / ((1 - 2 x) (1 - x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 18 2016 *)
LinearRecurrence[{2,1,-2},{2,-1,5},40] (* Harvey P. Dale, Feb 25 2022 *)

G.f.: (2-5*x+5*x^2)/((1-2*x)*(1-x^2)). - Vincenzo Librandi, Feb 18 2016

cp's OEIS Frontend

A269018 Primes p of the form 2^k + 2*(-1)^k - 1.

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