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%I #12 Jun 03 2019 01:11:29
%S 4,-6,6,-21,35,-35,21,-10,221,-77,55,-55,77,-221,10,-33,46513,-493,
%T 377,-119,187,-1333,559,-559,1333,-187,119,-377,493,-46513,33,-14,143,
%U -209,629,-14527,2881,-1189,533,-161,391,-15229,2449,-2263,3139,-1073,95,-95
%N The Euclid tree with root 1 encoded by semiprimes, read across levels.
%C The Euclid tree with root 1 is A295515 (sometimes called Calkin-Wilf tree).
%C For a positive rational r we use the Schinzel-Sierpiński encoding r -> [p, q] as described in A295511 and encode r as sgn*p*q where sgn is -1 if r < 1, else +1.
%C Apart from a(1) all terms are squarefree.
%D E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232.
%H N. Calkin and H. S. Wilf, <a href="https://www.jstor.org/stable/2589182">Recounting the rationals</a>, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.
%H Matthew M. Conroy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/CONROY/conroy.html">A sequence related to a conjecture of Schinzel</a>, J. Integ. Seqs. Vol. 4 (2001), #01.1.7.
%H P. D. T. A. Elliott, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002212722">The multiplicative group of rationals generated by the shifted primes. I.</a>, J. Reine Angew. Math. 463 (1995), 169-216.
%H P. D. T. A. Elliott, <a href="http://dx.doi.org/10.1515/crll.2000.017">The multiplicative group of rationals generated by the shifted primes. II.</a> J. Reine Angew. Math. 519 (2000), 59-71.
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SchinzelSierpinskiConjectureAndCalkinWilfTree">The Schinzel-Sierpiński conjecture and the Calkin-Wilf tree</a>.
%H A. Malter, D. Schleicher, D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/doi/10.4169/amer.math.monthly.120.03.243/NewLooksAtOldNumberTheory.pdf">New looks at old number theory</a>, Amer. Math. Monthly, 120(3), 2013, pp. 243-264.
%H A. Schinzel and W. Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa432.pdf">Sur certaines hypothèses concernant les nombres premiers</a>, Acta Arithmetica 4 (1958), 185-208; erratum 5 (1958) p. 259.
%e The tree starts:
%e 4
%e -6 6
%e -21 35 -35 21
%e -10 221 -77 55 -55 77 -221 10
%p EuclidTree := proc(n) local k, DijkstraFusc;
%p DijkstraFusc := proc(m) option remember; local a, b, n; a := 1; b := 0; n := m;
%p while n > 0 do if type(n, odd) then b := a+b else a := a+b fi; n := iquo(n,2) od; b end:
%p seq(DijkstraFusc(k)/DijkstraFusc(k+1), k=2^(n-1)..2^n-1) end:
%p SchinzelSierpinski := proc(l) local a, b, r, p, q, sgn;
%p a := numer(l); b := denom(l); q := 2; sgn := `if`(a < b, -1, 1);
%p while q < 1000000000 do r := a*(q - 1); if r mod b = 0 then p := r/b + 1;
%p if isprime(p) then return(sgn*p*q) fi fi; q := nextprime(q); od;
%p print("Search limit reached!", a, b) end:
%p Tree := level -> seq(SchinzelSierpinski(l), l=EuclidTree(level)): seq(Tree(n), n=1..6);
%Y Cf. A294442, A295511, A295515.
%K sign,tabf
%O 1,1
%A _Peter Luschny_, Nov 23 2017
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