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A289772 a(n) is the numerator of b(n) where b(n) = 1/(3*(1+2*A112765(n) - b(n-1))) and b(0) = 0, where A112765(n) is the 5-adic valuation of n.

%I A289772 #24 Dec 12 2023 13:42:14
%S A289772 0,1,1,2,1,1,2,5,3,4,1,5,4,7,3,2,3,7,4,5,1,4,3,5,2,1,3,8,5,7,2,11,9,
%T A289772 16,7,5,8,19,11,14,3,13,10,17,7,4,5,11,6,7,1,8,7,13,6,5,9,22,13,17,4,
%U A289772 19,15,26,11,7,10,23,13,16,3,11,8,13,5,2,5,13,8,11,3
%N A289772 a(n) is the numerator of b(n) where b(n) = 1/(3*(1+2*A112765(n) - b(n-1))) and b(0) = 0, where A112765(n) is the 5-adic valuation of n.
%C A289772 For n>0, a(n)/A289773(n) lists the rationals of a quinary analog of the Calkin-Wilf tree. See the Ponton link.
%H A289772 Robert Israel, <a href="/A289772/b289772.txt">Table of n, a(n) for n = 0..10000</a>
%H A289772 Lionel Ponton, <a href="https://arxiv.org/abs/1707.02366">Two trees enumerating the positive rationals</a>, arXiv:1707.02366 [math.NT], 2017. See p. 7.
%H A289772 Lionel Ponton, <a href="http://math.colgate.edu/~integers/sjs17/sjs17.Abstract.html">Two trees enumerating the positive rationals</a>, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A17.
%e A289772 Tree of rationals begin:
%e A289772 0;
%e A289772 1/3;
%e A289772 1/2, 2/3, 1, 1/6, 2/5;
%e A289772 5/9, 3/4, 4/3, 1/5, 5/12, 4/7, 7/9, 3/2, 2/9, 3/7, 7/12, 4/5, 5/3, 1/4, 4/9, 3/5, 5/6, 2, 1/9, 3/8, 8/15, 5/7, 7/6, 2/11, 11/27;
%e A289772 ...
%p A289772 b:= proc(n) option remember; 1/(3*(1+2*padic:-ordp(n,5)-procname(n-1))) end proc:
%p A289772 b(0):= 0:
%p A289772 map(numer@b, [$0..100]); # _Robert Israel_, Jul 12 2017
%t A289772 a[0] = 0; a[n_] := a[n] = 1/(3 (1 + 2 IntegerExponent[n, 5] - a[n - 1])); Table[Numerator@ a@ n, {n, 0, 80}] (* _Michael De Vlieger_, Jul 12 2017 *)
%o A289772 (PARI) b(n) = if (n==0, 0, 1/(3*(1+2*valuation(n, 5) - b(n-1))));
%o A289772 lista(nn) = for (n=0, nn, print1(numerator(b(n)), ", "));
%Y A289772 Cf. A002487, A277749, A277750, A289773.
%K A289772 nonn,frac,tabf,look
%O A289772 0,4
%A A289772 _Michel Marcus_, Jul 12 2017