This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A295511 #26 Jun 03 2019 01:13:25
%S A295511 2,2,2,3,3,2,3,7,7,5,5,7,7,3,2,5,17,13,7,11,11,5,5,11,11,7,13,17,5,2,
%T A295511 3,11,241,193,17,29,29,13,7,17,17,11,31,43,43,13,13,43,43,31,11,17,17,
%U A295511 7,13,29,29,17,193,241,11,3
%N A295511 The Schinzel-Sierpiński tree of fractions, read across levels.
%C A295511 A conjecture of Schinzel and Sierpiński asserts that every positive rational number r can be represented as a quotient of shifted primes such that r = (p-1)/(q-1).
%C A295511 The function r -> [p, q] will be called the Schinzel-Sierpiński encoding of r if q is the smallest prime such that r = (p-1)/(q-1) for some prime p. In the case that no pair of such primes exists we set by convention p = q = 1.
%C A295511 The Schinzel-Sierpiński tree of fractions is the Euclid tree A295515 with root 1 and fractions represented by the Schinzel-Sierpiński encoding.
%D A295511 E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232.
%H A295511 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 A295511 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 A295511 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 A295511 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 A295511 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SchinzelSierpinskiConjectureAndCalkinWilfTree">The Schinzel-Sierpiński conjecture and the Calkin-Wilf tree</a>.
%H A295511 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 (2013), 243-264.
%H A295511 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 A295511 The tree starts:
%e A295511 2/2
%e A295511 2/3 3/2
%e A295511 3/7 7/5 5/7 7/3
%e A295511 2/5 17/13 7/11 11/5 5/11 11/7 13/17 5/2
%e A295511 .
%e A295511 The numerators of the terms written as an array (the denominators are given by reversion of the arrays):
%e A295511 1: 2
%e A295511 2: 2, 3
%e A295511 3: 3, 7, 5, 7
%e A295511 4: 2, 17, 7, 11, 5, 11, 13, 5
%e A295511 5: 3, 241, 17, 29, 7, 17, 31, 43, 13, 43, 11, 17, 13, 29, 193, 11
%o A295511 (Sage)
%o A295511 def EuclidTree(n): # with root 1
%o A295511 def DijkstraFusc(m):
%o A295511 a, b, k = 1, 0, m
%o A295511 while k > 0:
%o A295511 if k % 2 == 1: b += a
%o A295511 else: a += b
%o A295511 k = k >> 1
%o A295511 return b
%o A295511 DF = [DijkstraFusc(k) for k in (2^(n-1)..2^n)]
%o A295511 return [DF[j]/DF[j+1] for j in (0..2^(n-1)-1)]
%o A295511 def SchinzelSierpinski(l):
%o A295511 a, b = l.numerator(), l.denominator()
%o A295511 p, q = 1, 2
%o A295511 while q < 1000000000: # search limit
%o A295511 r = a*(q - 1)
%o A295511 if b.divides(r):
%o A295511 p = r // b + 1
%o A295511 if is_prime(p): return p/q
%o A295511 q = next_prime(q)
%o A295511 print("Search limit reached for ", l); return 0
%o A295511 def SSETree(level):
%o A295511 return [SchinzelSierpinski(l) for l in EuclidTree(level)]
%o A295511 # With the imperfection that Sage reduces 2/2 automatically to 1.
%o A295511 for level in (1..6): print(SSETree(level))
%Y A295511 Cf. A002487, A294442, A295510, A295512, A295515.
%K A295511 nonn,tabf
%O A295511 1,1
%A A295511 _Peter Luschny_, Nov 23 2017