A002716 An infinite coprime sequence defined by recursion. (Formerly M2488 N0986)

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

Offset

0,1

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

Table of n, a(n) for n=0..12.

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.

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

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_ *)

Prog

(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 */

Crossrefs

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

Sequence in context: A131020 A283398 A084706 * A046154 A180120 A075704

Adjacent sequences: A002713 A002714 A002715 * A002717 A002718 A002719

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

More terms from Jeffrey Shallit.

Edited by Michael Somos, Feb 01 2004

Status

approved