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A002716 An infinite coprime sequence defined by recursion.

%I A002716 M2488 N0986 #36 Oct 20 2023 09:23:04
%S A002716 3,5,13,17,241,257,65281,65537,4294901761,4294967297,
%T A002716 18446744069414584321,18446744073709551617,
%U A002716 340282366920938463444927863358058659841
%N A002716 An infinite coprime sequence defined by recursion.
%C A002716 Every term is relatively prime to all others. - _Michael Somos_, Feb 01 2004
%D A002716 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002716 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002716 A. W. F. Edwards, <a href="https://www.jstor.org/stable/3611702">Infinite coprime sequences</a>, Math. Gaz., 48 (1964), 416-422.
%H A002716 A. W. F. Edwards, <a href="/A002715/a002715.pdf">Infinite coprime sequences</a>, Math. Gaz., 48 (1964), 416-422. [Annotated scanned copy]
%H A002716 R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.
%F A002716 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
%F A002716 Conjecture: a(2n+1)=A001146(n+1)+1. - _R. J. Mathar_, May 15 2007
%F A002716 a(2*n) = A220294(n). a(2*n + 1) = A000215(n+1). - _Michael Somos_, Dec 10 2012
%t A002716 a[0] = 3; a[1] = 5;
%t A002716 a[n_] := a[n] = If[OddQ[n], a[n-1] + a[n-2] - 1, a[n-1]^2 - 3*a[n-1] + 3];
%t A002716 Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Aug 16 2018, after _Michel Somos_ *)
%o A002716 (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 */
%Y A002716 Cf. A000215, A001685, A002715, A003686, A064526, A220294.
%K A002716 nonn
%O A002716 0,1
%A A002716 _N. J. A. Sloane_
%E A002716 More terms from _Jeffrey Shallit_
%E A002716 Edited by _Michael Somos_, Feb 01 2004