A295512 The Euclid tree with root 1 encoded by semiprimes, read across levels.

4, -6, 6, -21, 35, -35, 21, -10, 221, -77, 55, -55, 77, -221, 10, -33, 46513, -493, 377, -119, 187, -1333, 559, -559, 1333, -187, 119, -377, 493, -46513, 33, -14, 143, -209, 629, -14527, 2881, -1189, 533, -161, 391, -15229, 2449, -2263, 3139, -1073, 95, -95

Offset

1,1

Comments

The Euclid tree with root 1 is A295515 (sometimes called Calkin-Wilf tree).

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.

Apart from a(1) all terms are squarefree.

References

E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232.

Links

Table of n, a(n) for n=1..48.

N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.

Matthew M. Conroy, A sequence related to a conjecture of Schinzel, J. Integ. Seqs. Vol. 4 (2001), #01.1.7.

P. D. T. A. Elliott, The multiplicative group of rationals generated by the shifted primes. I., J. Reine Angew. Math. 463 (1995), 169-216.

P. D. T. A. Elliott, The multiplicative group of rationals generated by the shifted primes. II. J. Reine Angew. Math. 519 (2000), 59-71.

Peter Luschny, The Schinzel-Sierpiński conjecture and the Calkin-Wilf tree.

A. Malter, D. Schleicher, D. Zagier, New looks at old number theory, Amer. Math. Monthly, 120(3), 2013, pp. 243-264.

A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arithmetica 4 (1958), 185-208; erratum 5 (1958) p. 259.

Example

The tree starts:
                     4
            -6                 6
       -21       35      -35       21
    -10  221  -77  55  -55  77  -221  10

Maple

EuclidTree := proc(n) local k, DijkstraFusc;
DijkstraFusc := proc(m) option remember; local a, b, n; a := 1; b := 0; n := m;
while n > 0 do if type(n, odd) then b := a+b else a := a+b fi; n := iquo(n, 2) od; b end:
seq(DijkstraFusc(k)/DijkstraFusc(k+1), k=2^(n-1)..2^n-1) end:
SchinzelSierpinski := proc(l) local a, b, r, p, q, sgn;
a := numer(l); b := denom(l); q := 2; sgn := `if`(a < b, -1, 1);
while q < 1000000000 do r := a*(q - 1); if r mod b = 0 then p := r/b + 1;
if isprime(p) then return(sgn*p*q) fi fi; q := nextprime(q); od;
print("Search limit reached!", a, b) end:
Tree := level -> seq(SchinzelSierpinski(l), l=EuclidTree(level)): seq(Tree(n), n=1..6);

Crossrefs

Cf. A294442, A295511, A295515.

Sequence in context: A351649 A019090 A145692 * A064214 A019203 A365084

Adjacent sequences: A295509 A295510 A295511 * A295513 A295514 A295515

Keyword

sign,tabf

Author

Peter Luschny, Nov 23 2017

Status

approved