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%I #28 Nov 25 2015 21:49:30
%S 1,3,5,7,2,9,11,13,4,15,17,19,6,8,21,23,10,12,25,14,27,29,31,16,33,18,
%T 35,37,20,39,41,43,22,45,47,24,49,26,51,53,28,55,30,32,57,59,61,34,63,
%U 36,65,67,38,69,71,73,40,42,44,75,46,77,79,48,50
%N "Shuffled bisection" of positive even and odd numbers, Type 2 (see "Comments" for rules generating the sequence).
%C All terms are positive, with even and odd terms appearing in due course in order. One might think of the sequence as the result of two poorly-shuffled decks of cards with numbers in increasing order -- one deck with all the positive even numbers and one with all the positive odd numbers, following this set of rules:
%C Start with a(1)=1; take next odd number until preceding an odd composite, then take next even number. Take next odd unless coprime to last term, otherwise take next even. Repeat indefinitely.
%C Odd terms precede primes (by definition); primes are followed by even terms unless smallest twin prime.
%C Excluding a(1)..a(4), the maximum length of odd strings is 3. Is there a limit to the maximum length of even strings?
%C Same as A263411(n), n<=25.
%C If seeds are a(1)=1 and a(2)=2, the sequences converge after a(17).
%H Michael De Vlieger, <a href="/A263792/b263792.txt">Table of n, a(n) for n = 1..10000</a>
%e a(11)=17; since next odd is prime, take a(12)=19. Since next odd is composite, take next even, so a(13)=6. Since next odd (21) is not coprime to 6, take a(14)=8. Since 21 is coprime to 8, take a(15)=21.
%t f[n_] := Block[{a = 3, b = 2, t = {1}, k}, Do[Which[And[OddQ@ t[[k - 1]], PrimeQ@ a], AppendTo[t, a]; a += 2, And[OddQ@ t[[k - 1]], CompositeQ@ a], AppendTo[t, b]; b += 2, And[EvenQ@ t[[k - 1]], CoprimeQ[a, t[[k - 1]]]], AppendTo[t, a]; a += 2, And[EvenQ@ t[[k - 1]], ! CoprimeQ[a, t[[k - 1]]]], AppendTo[t, b]; b += 2, True, AppendTo[t, 1]], {k, 2, n}]; t]; f@ 120 (* _Michael De Vlieger_, Oct 27 2015 *)
%Y Cf. A005408, A005843, A263411.
%K nonn
%O 1,2
%A _Bob Selcoe_, Oct 26 2015
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