A111800 - OEIS

%I #14 Nov 11 2015 03:45:22

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

%C A061396(n) gives the number of times that 2n+1 appears in this sequence.

%H Alois P. Heinz, <a href="/A111800/b111800.txt">Table of n, a(n) for n = 1..10000</a>

%H J. Awbrey, <a href="/A061396/a061396a.txt">Illustrations of Rotes for Small Integers</a>

%H J. Awbrey, <a href="https://oeis.org/wiki/Riffs_and_Rotes">Riffs and Rotes</a>

%F a(Prod(p_i^e_i)) = 1 + Sum(a(i) + a(e_i)), product over nonzero e_i in prime factorization of n.

%e Writing prime(i)^j as i:j and using equal signs between identified nodes:

%e 2500 = 4 * 625 = 2^2 5^4 = 1:2 3:4 has the following rote:

%e ` ` ` ` ` ` ` `

%e ` ` ` o-o ` o-o

%e ` ` ` | ` ` | `

%e ` o-o o-o o-o `

%e ` | ` | ` | ` `

%e o-o ` o---o ` `

%e | ` ` | ` ` ` `

%e O=====O ` ` ` `

%e ` ` ` ` ` ` ` `

%e So a(2500) = a(1:2 3:4) = a(1)+a(2)+a(3)+a(4)+1 = 1+3+5+5+1 = 15.

%p with(numtheory):

%p a:= proc(n) option remember;

%p       1+add(a(pi(i[1]))+a(i[2]), i=ifactors(n)[2])

%p     end:

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 25 2015

%t 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 Cf. A061396, A062504, A062537, A062860, A106177, A109300, A109301.

%K nonn

%O 1,2

%A Jon Awbrey, Aug 17 2005, based on calculations by _David W. Wilson_