A002716 An infinite coprime sequence defined by recursion.

3, 5, 13, 17, 241, 257, 65281, 65537, 4294901761, 4294967297, 18446744069414584321, 18446744073709551617, 340282366920938463444927863358058659841

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Every term is relatively prime to all others. - Michael Somos, Feb 01 2004

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422.
• A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422. [Annotated scanned copy]
• R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.

Crossrefs

Cf. A000215, A001685, A002715, A003686, A064526, A220294.

Programs

Mathematica

a[0] = 3; a[1] = 5;
a[n_] := a[n] = If[OddQ[n], a[n-1] + a[n-2] - 1, a[n-1]^2 - 3*a[n-1] + 3];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 16 2018, after Michel Somos *)

PARI

{a(n) = if( n<2, 3 * (n>=0) + 2 * (n>0), if( n%2, a(n-1) + a(n-2) - 1, a(n-1)^2 - 3 * a(n-1) + 3))} /* Michael Somos, Feb 01 2004 */

Formula

a(2*n + 1) = a(2*n) + a(2*n - 1) - 1, a(2*n) = a(2*n - 1)^2 - 3 * a(2*n - 1) + 3, a(0) = 3, a(1) = 5. - Michael Somos, Feb 01 2004

Conjecture: a(2n+1)=A001146(n+1)+1. - R. J. Mathar, May 15 2007

a(2*n) = A220294(n). a(2*n + 1) = A000215(n+1). - Michael Somos, Dec 10 2012

Extensions

More terms from Jeffrey Shallit

Edited by Michael Somos, Feb 01 2004