A289773 - OEIS

%I #21 Dec 12 2023 13:41:46

%S 1,3,2,3,1,6,5,9,4,3,5,12,7,9,2,9,7,12,5,3,4,9,5,6,1,9,8,15,7,6,11,27,

%T 16,21,5,24,19,33,14,9,13,30,17,21,4,15,11,18,7,3,8,21,13,18,5,27,22,

%U 39,17,12,19,45,26,33,7,30,23,39,16,9,11,24,13,15,2,15,13

%N a(n) is the denominator of b(n) where b(n) = 1/(3*(1+2*A112765(n) - b(n-1))) and b(0) = 0, with A112765(n) being the 5-adic valuation of n.

%C For n>0, A289772(n)/a(n) lists the rationals of a quinary analog of the Calkin-Wilf tree. See the Ponton link.

%H Robert Israel, <a href="/A289773/b289773.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 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 Tree of rationals begin:

%e 0;

%e 1/3;

%e 1/2, 2/3, 1, 1/6, 2/5;

%e 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 ...

%p b:= proc(n) option remember; 1/(3*(1+2*padic:-ordp(n,5)-procname(n-1))) end proc:

%p b(0):= 0:

%p map(denom@b, [$0..100]); # _Robert Israel_, Jul 12 2017

%t a[0] = 0; a[n_] := a[n] = 1/(3 (1 + 2 IntegerExponent[n, 5] - a[n - 1])); Table[Denominator@ a@ n, {n, 0, 76}] (* _Michael De Vlieger_, Jul 12 2017 *)

%o (PARI) b(n) = if (n==0, 0, 1/(3*(1+2*valuation(n, 5) - b(n-1))));

%o lista(nn) = for (n=0, nn, print1(denominator(b(n)), ", "));

%Y Cf. A002487, A277749, A277750, A289772.

%K nonn,frac,tabf,look

%O 0,2

%A _Michel Marcus_, Jul 12 2017