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

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, 11, 9, 13, 11, 13, 11, 11, 11, 13, 13, 11, 13, 11, 13, 13, 11, 13, 11, 9, 11, 13, 13, 9, 11, 15, 13, 13, 13, 11, 15, 11, 13, 13, 9, 15, 15, 11, 13, 13, 15, 13, 13, 13, 13, 13, 13, 15, 15

Offset: 1

Author

Jon Awbrey, Aug 17 2005, based on calculations by David W. Wilson

Keywords

• nonn
• changed

Comments

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

Examples

			Writing prime(i)^j as i:j and using equal signs between identified nodes:
2500 = 4 * 625 = 2^2 5^4 = 1:2 3:4 has the following rote:
  ` ` ` ` ` ` ` `
  ` ` ` o-o ` o-o
  ` ` ` | ` ` | `
  ` o-o o-o o-o `
  ` | ` | ` | ` `
  o-o ` o---o ` `
  | ` ` | ` ` ` `
  O=====O ` ` ` `
  ` ` ` ` ` ` ` `
So a(2500) = a(1:2 3:4) = a(1)+a(2)+a(3)+a(4)+1 = 1+3+5+5+1 = 15.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• J. Awbrey, Illustrations of Rotes for Small Integers
• J. Awbrey, Riffs and Rotes

Crossrefs

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

Programs

Maple

with(numtheory):
a:= proc(n) option remember;
      1+add(a(pi(i[1]))+a(i[2]), i=ifactors(n)[2])
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 25 2015

Mathematica

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 *)

Formula

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