A130730 -id:A130730 - cp's OEIS frontend

A063486 a(n) = 2^(2^n) + 5.

7, 9, 21, 261, 65541, 4294967301, 18446744073709551621, 340282366920938463463374607431768211461, 115792089237316195423570985008687907853269984665640564039457584007913129639941

Offset: 0

Jason Earls, Jul 28 2001

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 238.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Example 5.1 on page 153.

Harry J. Smith, Table of n, a(n) for n=0..11
Tigran Hakobyan, On the unboundedness of common divisors of distinct terms of the sequence a(n)=2^2^n+d for d>1, arXiv:1601.04946 [math.NT], 2016.

Cf. A000215, A001146, A130729, A130730.

2^2^Range[0, 10] + 5 (* Paolo Xausa, Apr 17 2024 *)

PARI
```
for(n=0,8,print1(2^(2^n)+5, ", "))
```

{ for (n=0, 11, write("b063486.txt", n, " ", 2^(2^n) + 5) ) } \\ Harry J. Smith, Aug 23 2009

[2**2**n + 5 for n in (0..8)] # Stefano Spezia, Jul 20 2025

A130729 Fermat numbers of order 3 or F(n,3) = 2^(2^n)+3.

Original entry on oeis.org

5, 7, 19, 259, 65539, 4294967299, 18446744073709551619, 340282366920938463463374607431768211459, 115792089237316195423570985008687907853269984665640564039457584007913129639939

Offset: 0

Cino Hilliard, Jul 05 2007

nonn

This is equivalent to F(n)+2 or 2^(2^n)+ 1 + 2. Conjecture: If n is odd, 7 is a divisor of F(n,3).

The conjecture is true: the order of 2 mod 7 is 3, and if n is odd then 2^n == 2 mod 3 so 2^(2^n) + 3 == 2^2 + 3 == 0 mod 7. - Robert Israel, Nov 20 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..11
Tigran Hakobyan, On the unboundedness of common divisors of distinct terms of the sequence a(n)=2^2^n+d for d>1, arXiv:1601.04946 [math.NT], 2016.

Cf. A063486, A130730.

[2^(2^n) + 3: n in [0..11]]; // Vincenzo Librandi, Jan 09 2013

Table[(2^(2^n) + 3), {n, 0, 15}] (* Vincenzo Librandi, Jan 09 2013 *)

fplusm(n,m)= { local(x,y); for(x=0,n, y=2^(2^x)+m; print1(y",") ) }

F(n,m): The n-th Fermat number of order m = 2^(2^n)+ m. The traditional Fermat numbers are F(n,1) or Fermat numbers of order 1.

A274022 Primes of the form 2^(2^k) + 3.

Original entry on oeis.org

5, 7, 19, 65539

Offset: 1

Vincenzo Librandi, Jun 07 2016

nonn

Terms given correspond to n = 0, 1, 2, and 4.

Next term >= 2^2^28 + 3. - Charles R Greathouse IV, Jun 08 2016

Cf. A019434, A130730.

Cf. similar sequences listed in A273547.

[a: n in [0..10] | IsPrime(a) where a is 2^(2^n)+3];

Select[Table[2^(2^n) + 3, {n, 0, 15}], PrimeQ]

for(n=0,4, if(ispseudoprime(t=2^2^n+3), print1(t", "))) \\ Charles R Greathouse IV, Jun 08 2016

A160021 a(n)=2^(2^n)+33, Fermat numbers of order 33.

Original entry on oeis.org

35, 37, 49, 289, 65569, 4294967329, 18446744073709551649, 340282366920938463463374607431768211489, 115792089237316195423570985008687907853269984665640564039457584007913129639969

Offset: 1

Cino Hilliard, Apr 30 2009

nonn

Fermat numbers of order m are defined by F(n,m) = 2^(2^n)+m = A001146(n)+m.

F(1,33) = 37 is the only prime in this sequence. (If n is even, 7 divides F(n,33). For n > 2, 17 divides F(n,33). Proofs are in the link.)

Vincenzo Librandi, Table of n, a(n) for n = 1..12
Cino Hilliard, Fermat numbers of order m

Cf. A130730 (order 7).

[2^(2^n)+33: n in [0..11]]; // Vincenzo Librandi, Jan 09 2013

Table[(2^(2^n) + 33), {n, 0, 15}] (* Vincenzo Librandi, Jan 09 2013 *)

g(n) = for(x=0,n,y=2^(2^x)+33;print1(y","))

Edited by R. J. Mathar, May 08 2009

A274023 Primes of the form 2^(2^k) + 13.

Original entry on oeis.org

17, 29, 269, 18446744073709551629

Offset: 1

Vincenzo Librandi, Jun 07 2016

nonn

Terms given correspond to k = 1, 2, 3 and 6.

Next term >= 2^2^39 + 13. - Charles R Greathouse IV, Jun 08 2016

Cf. A019434, A104067, A130730, A274022.

Cf. similar sequences listed in A273547.

[a: n in [0..10] | IsPrime(a) where a is 2^(2^n)+13];

Select[Table[2^(2^n) + 13, {n, 0, 15}], PrimeQ]

for(n=1,6, if(ispseudoprime(t=2^2^n+13), print1(t", "))) \\ Charles R Greathouse IV, Jun 08 2016

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A063486 a(n) = 2^(2^n) + 5.

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A130729 Fermat numbers of order 3 or F(n,3) = 2^(2^n)+3.

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A274022 Primes of the form 2^(2^k) + 3.

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A160021 a(n)=2^(2^n)+33, Fermat numbers of order 33.

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A274023 Primes of the form 2^(2^k) + 13.

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