A070969 -id:A070969 - cp's OEIS frontend

A220161 a(n) = 1 + 2^(2^n) + 2^(2^(n+1)).

7, 21, 273, 65793, 4295032833, 18446744078004518913, 340282366920938463481821351505477763073, 115792089237316195423570985008687907853610267032561502502920958615344897851393

Offset: 0

Michel Marcus, Dec 06 2012

nonn

For n >= 1, W. Sierpiński proves that a(n) is divisible by 21.

For n >= 1, A. Engel shows that a(n) = a(n-1) * A220294(n-1). - Hans Havermann, Mar 07 2015

Arthur Engel, Problem-Solving Strategies, Springer, 1998, pages 121-122 (E3, said to be a "recent competition problem from the former USSR").
W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #123.

G. C. Greubel, Table of n, a(n) for n = 0..10
K. Zelator, On a theorem on sums of the form 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m) and a result linking Fermat with Mersenne numbers, arXiv:0806.1514 [math.GM], 2008.

Cf. A000215, A002061, A070969, A087046, A220294, A255770, A255771, A255772.

[1 + 2^(2^n) + 2^(2^(n+1)): n in [0..10]]; // G. C. Greubel, Aug 10 2018

Table[1+2^(2^n)+2^(2^(n+1)),{n,0,7}] (* Harvey P. Dale, Dec 16 2015 *)

A220161(n):=1 + 2^(2^n) + 2^(2^(n+1))$ makelist(A220161(n),n,0,10); /* Martin Ettl, Dec 10 2012 */

vector(10, n, n--; 1 + 2^(2^n) + 2^(2^(n+1))) \\ G. C. Greubel, Aug 10 2018

def a(n): return 1 + 2**(2**n) + 2**(2**(n+1))
print([a(n) for n in range(8)]) # Michael S. Branicky, Jul 21 2021

a(n) = A000215(n+1) + A000215(n) - 1.
A070969(n) = sqrt(4*a(n) - 3). a(n+1) = a(n) * (1 + a(n) - A070969(n)) = a(n) * (1 + A087046(n+2)) hence a(n) divides a(n+1). - Michael Somos, Dec 10 2012
a(n) = A002061(A000215(n)). - Pontus von Brömssen, Aug 31 2021

A006485 a(n) = (2^(2^n + 1) + 1)/3.

Original entry on oeis.org

3, 11, 171, 43691, 2863311531, 12297829382473034411, 226854911280625642308916404954512140971, 77194726158210796949047323339125271902179989777093709359638389338608753093291

Offset: 1

Dennis S. Kluk (mathemagician(AT)ameritech.net)

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Delbert L. Johnson, Table of n, a(n) for n = 1..11
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979.

Cf. A001045, A070969.

lst={};Do[AppendTo[lst, (2^(2^n+1)+1)/3], {n, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
Table[(1+2^(2^n+1))/3,{n,10}] (* Harvey P. Dale, Apr 03 2023 *)

{a(n) = if(n > 0, (2^(2^n + 1) + 1)/3, 0)}; /* Michael Somos, Mar 30 2020 */

a(n) = A001045(2^n+1) = (3*a(n-1)^2 + 1)/2 - a(n-1). - Michael Somos, Mar 30 2020

a(n) = A070969(n)/3. - Alois P. Heinz, Mar 28 2023

A109384 a(n) = 2^(2^(n+2)+1).

Original entry on oeis.org

32, 512, 131072, 8589934592, 36893488147419103232, 680564733841876926926749214863536422912, 231584178474632390847141970017375815706539969331281128078915168015826259279872

Offset: 0

Creighton Dement, Sep 02 2005

Cf. A070969.

[2^(2^(n+2)+1) : n in [0..7]]; // Wesley Ivan Hurt, Apr 07 2015

A109384:=n->2^(2^(n+2)+1): seq(A109384(n), n=0..7); # Wesley Ivan Hurt, Apr 07 2015

Table[2^(2^(n + 2) + 1), {n, 0, 7}] (* Wesley Ivan Hurt, Apr 07 2015 *)

a(n)=2^(2^(n+2)+1) \\ Charles R Greathouse IV, Apr 07 2015

a(n) = A070969(n+2)-1.

A101329 2^(2^(2^x)+1)-1 = M(F(x)) == Mersenne(Fermat(x)).

Original entry on oeis.org

7, 31, 131071, 231584178474632390847141970017375815706539969331281128078915168015826259279871

Offset: 0

Jorge Coveiro, Dec 24 2004

nonn

Mersenne-Fermat numbers.

			M(F(0))=7
M(F(1))=31
M(F(2))=131071
M(F(3))=231584178474632390847141970017375815706539969331281128078915168015826259279871

PARI
```
for(x=0,3,print(2^(2^(2^x)+1)-1))
```

a(n) = A070969(2^n)-2. - R. J. Mathar, Apr 22 2007

cp's OEIS Frontend

A220161 a(n) = 1 + 2^(2^n) + 2^(2^(n+1)).

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A006485 a(n) = (2^(2^n + 1) + 1)/3.

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A109384 a(n) = 2^(2^(n+2)+1).

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A101329 2^(2^(2^x)+1)-1 = M(F(x)) == Mersenne(Fermat(x)).

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