cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A220294 a(n) = 1 - 2^(2^n) + 2^(2^(n+1)).

Original entry on oeis.org

3, 13, 241, 65281, 4294901761, 18446744069414584321, 340282366920938463444927863358058659841, 115792089237316195423570985008687907852929702298719625575994209400481361428481
Offset: 0

Views

Author

Michael Somos, Dec 10 2012

Keywords

Comments

An infinite coprime sequence defined by recursion.

Crossrefs

Programs

  • Magma
    [1 - 2^(2^n) + 2^(2^(n+1)): n in [0..10]]; // G. C. Greubel, Aug 10 2018
  • Mathematica
    Table[4^(2^m) - 2^(2^m) + 1, {m, 0, 7}] (* Michael De Vlieger, Aug 02 2016 *)
  • Maxima
    A220294(n):=1 - 2^(2^n) + 2^(2^(n+1))$ makelist(A220294(n),n,0,10); /* Martin Ettl, Dec 10 2012 */
    
  • PARI
    {a(n) = if( n<0, 0, 1 - 2^(2^n) + 2^(2^(n+1)))};
    

Formula

A220161(n+1) = a(n) * A220161(n).
a(n+1) = 1 + (a(n) - 1) * (A220161(n) - 1).
a(n) = A002716(2*n) = 1 + A087046(n+2) = 1 + A111403(n).
a(n) = A002061(A001146(n)). - Pontus von Brömssen, Aug 31 2021

A255772 Start with 7; thereafter, in order of appearance, the prime factors of A220294.

Original entry on oeis.org

7, 3, 13, 241, 97, 673, 193, 22253377, 18446744069414584321, 769, 442499826945303593556473164314770689, 349621839326921795694385454593, 331192380488114152600457428497953408512758882817, 212780015855109121
Offset: 1

Views

Author

Hans Havermann, Mar 06 2015

Keywords

Comments

A220161(n-1) = the product of the first A255770(n) terms.

References

  • Arthur Engel, Problem-Solving Strategies, Springer, 1998, pages 121-122 (E3, said to be a "recent competition problem from the former USSR").

Crossrefs

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

Original entry on oeis.org

7, 21, 273, 65793, 4295032833, 18446744078004518913, 340282366920938463481821351505477763073, 115792089237316195423570985008687907853610267032561502502920958615344897851393
Offset: 0

Views

Author

Michel Marcus, Dec 06 2012

Keywords

Comments

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

References

  • 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.

Crossrefs

Programs

  • Magma
    [1 + 2^(2^n) + 2^(2^(n+1)): n in [0..10]]; // G. C. Greubel, Aug 10 2018
    
  • Mathematica
    Table[1+2^(2^n)+2^(2^(n+1)),{n,0,7}] (* Harvey P. Dale, Dec 16 2015 *)
  • Maxima
    A220161(n):=1 + 2^(2^n) + 2^(2^(n+1))$ makelist(A220161(n),n,0,10); /* Martin Ettl, Dec 10 2012 */
    
  • PARI
    vector(10, n, n--; 1 + 2^(2^n) + 2^(2^(n+1))) \\ G. C. Greubel, Aug 10 2018
    
  • Python
    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

Formula

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

A255771 Number of distinct prime factors of A220294(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 4, 2, 2
Offset: 0

Views

Author

Hans Havermann, Mar 06 2015

Keywords

Comments

These are the first differences of A255770.

Examples

			A220294(0) = 3 so a(0) = 1.
A220294(1) = 13 so a(1) = 1.
A220294(2) = 241 so a(2) = 1.
A220294(3) = 97*673 so a(3) = 2.
A220294(4) = 193*22253377 so a(4) = 2.
		

References

  • Arthur Engel, Problem-Solving Strategies, Springer, 1998, pages 121-122 (E3, said to be a "recent competition problem from the former USSR").

Crossrefs

Extensions

Offset changed by Arkadiusz Wesolowski, Aug 01 2016
a(9) was found in 2008 by Geoffrey Reynolds. a(10) was found by Anders Björn and Hans Riesel. - Arkadiusz Wesolowski, Aug 02 2016
Showing 1-4 of 4 results.