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%I #22 Oct 28 2021 03:58:48
%S 2,4,6,3,5,7,11,8,9,10,13,17,19,23,29,31,12,14,15,16,18,37,41,43,47,
%T 53,59,61,67,20,21,22,24,25,26,27,71,73,79,83,89,97,101,103,107,28,30,
%U 32,33,34,35,36,38,39,40,42
%N Alternating runs of primes and composites, with the runs of primes being of composite length and the runs of composites being of prime length.
%C In other words, use sequence A073846 to list alternating runs of primes and composites, with the number of elements in each run given by successive terms in A073846 - with each even-indexed term of A073846 (being itself prime) denoting the length of each run of composites and each odd-indexed term of A073846 (being itself composite) denoting the length of each run of primes.
%e a(1) = 2, this being a length 1 (1 is initial index) run of primes.
%e a(2) = 4 & a(3) = 6, 4 and 6 being a length 2 (2 is first prime) run of composites.
%e a(4) = 3, a(5) = 5, a(6) = 7, and a(7) = 11 being a length 4 (4 is first composite) run of primes.
%e a(8) = 8, a(9) = 9, and a(10) = 10, being a length 3 (3 is 2nd prime) run of composites.
%t m=10;c1=Select[Range@m,!PrimeQ@#&];p1=Prime@Range@Total@c1;p2=Prime@Range@m;c2=Select[Range[2,2Total@p2],!PrimeQ@#&][[;;Total@p2]];t1=TakeList[p1,c1];t2=TakeList[c2,p2];min=Min[Length/@{t1,t2}];Flatten@Riffle[t1[[;;min]],t2[[;;min]]] (* _Giorgos Kalogeropoulos_, Jul 30 2021 *)
%Y Cf. A000040 (primes), A002808 (composites), A073846.
%K nonn,easy
%O 1,1
%A _Walter Carlini_, Jul 07 2021
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