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%I A295512 #12 Jun 03 2019 01:11:29
%S A295512 4,-6,6,-21,35,-35,21,-10,221,-77,55,-55,77,-221,10,-33,46513,-493,
%T A295512 377,-119,187,-1333,559,-559,1333,-187,119,-377,493,-46513,33,-14,143,
%U A295512 -209,629,-14527,2881,-1189,533,-161,391,-15229,2449,-2263,3139,-1073,95,-95
%N A295512 The Euclid tree with root 1 encoded by semiprimes, read across levels.
%C A295512 The Euclid tree with root 1 is A295515 (sometimes called Calkin-Wilf tree).
%C A295512 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 A295512 Apart from a(1) all terms are squarefree.
%D A295512 E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232.
%H A295512 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 A295512 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 A295512 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 A295512 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 A295512 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SchinzelSierpinskiConjectureAndCalkinWilfTree">The Schinzel-Sierpiński conjecture and the Calkin-Wilf tree</a>.
%H A295512 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 A295512 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 A295512 The tree starts:
%e A295512                      4
%e A295512             -6                 6
%e A295512        -21       35      -35       21
%e A295512     -10  221  -77  55  -55  77  -221  10
%p A295512 EuclidTree := proc(n) local k, DijkstraFusc;
%p A295512 DijkstraFusc := proc(m) option remember; local a, b, n; a := 1; b := 0; n := m;
%p A295512 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 A295512 seq(DijkstraFusc(k)/DijkstraFusc(k+1), k=2^(n-1)..2^n-1) end:
%p A295512 SchinzelSierpinski := proc(l) local a, b, r, p, q, sgn;
%p A295512 a := numer(l); b := denom(l); q := 2; sgn := `if`(a < b, -1, 1);
%p A295512 while q < 1000000000 do r := a*(q - 1); if r mod b = 0 then p := r/b + 1;
%p A295512 if isprime(p) then return(sgn*p*q) fi fi; q := nextprime(q); od;
%p A295512 print("Search limit reached!", a, b) end:
%p A295512 Tree := level -> seq(SchinzelSierpinski(l), l=EuclidTree(level)): seq(Tree(n), n=1..6);
%Y A295512 Cf. A294442, A295511, A295515.
%K A295512 sign,tabf
%O A295512 1,1
%A A295512 _Peter Luschny_, Nov 23 2017

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