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A111800 Order of the rote (rooted odd tree with only exponent symmetries) for n.

%I A111800 #17 Sep 02 2025 23:31:34
%S A111800 1,3,5,5,7,7,7,7,7,9,9,9,9,9,11,7,9,9,9,11,11,11,9,11,9,11,9,11,11,13,
%T A111800 11,9,13,11,13,11,11,11,13,13,11,13,11,13,13,11,13,11,9,11,13,13,9,11,
%U A111800 15,13,13,13,11,15,11,13,13,9,15,15,11,13,13,15,13,13,13,13,13,13,15,15
%N A111800 Order of the rote (rooted odd tree with only exponent symmetries) for n.
%C A111800 A061396(n) gives the number of times that 2n+1 appears in this sequence.
%H A111800 Alois P. Heinz, <a href="/A111800/b111800.txt">Table of n, a(n) for n = 1..10000</a>
%H A111800 J. Awbrey, <a href="/A061396/a061396a.txt">Illustrations of Rotes for Small Integers</a>
%H A111800 J. Awbrey, <a href="https://oeis.org/wiki/Riffs_and_Rotes">Riffs and Rotes</a>
%F A111800 a(Prod(p_i^e_i)) = 1 + Sum(a(i) + a(e_i)), product over nonzero e_i in prime factorization of n.
%e A111800 Writing prime(i)^j as i:j and using equal signs between identified nodes:
%e A111800 2500 = 4 * 625 = 2^2 5^4 = 1:2 3:4 has the following rote:
%e A111800   ` ` ` ` ` ` ` `
%e A111800   ` ` ` o-o ` o-o
%e A111800   ` ` ` | ` ` | `
%e A111800   ` o-o o-o o-o `
%e A111800   ` | ` | ` | ` `
%e A111800   o-o ` o---o ` `
%e A111800   | ` ` | ` ` ` `
%e A111800   O=====O ` ` ` `
%e A111800   ` ` ` ` ` ` ` `
%e A111800 So a(2500) = a(1:2 3:4) = a(1)+a(2)+a(3)+a(4)+1 = 1+3+5+5+1 = 15.
%p A111800 with(numtheory):
%p A111800 a:= proc(n) option remember;
%p A111800       1+add(a(pi(i[1]))+a(i[2]), i=ifactors(n)[2])
%p A111800     end:
%p A111800 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 25 2015
%t A111800 a[1] = 1; a[n_] := a[n] = 1+Sum[a[PrimePi[i[[1]] ] ] + a[i[[2]] ], {i, FactorInteger[n]}]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Nov 11 2015, after _Alois P. Heinz_ *)
%Y A111800 Cf. A061396, A062504, A062537, A062860, A106177, A109300, A109301.
%K A111800 nonn,changed
%O A111800 1,2
%A A111800 _Jon Awbrey_, Aug 17 2005, based on calculations by _David W. Wilson_