Riffs & Rotes & A061396 & A062504?

[David W. Wilson]

Jon Awbrey wrote:
> 
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
> 
> Re:
> 
> Notes on A062504 [pending]
> 
> Table 3.  Triangle in which k-th row lists natural number
>           values for the collection of riffs with k nodes:
> --o------------------------------------------------------------------------------
> k | natural numbers n such that |riff(n)| = k
> --o------------------------------------------------------------------------------
> 0 |    1;
> 1 |    2;
> 2 |    3,    4;
> 3 |    5,    6,    7,    8,     9,    16;
> 4 |   10,   11,   12,   13,    14,    17,     18,     19,     23,     25,
>   |   27,   32,   49,   53,    64,    81,    128,    256,    512,  65536;
> 5 |   15,   20,   21,   22,    24,    26,     28,     29,     31,     34,
>   |   36,   37,   38,   41,    43,    46,     48,     50,     54,     59,
>   |   61,   67,   83,   97,    98,   103,    106,    121,    125,    131,
>   |  162,  169,  227,  241,   243,   289,    311,    343,    361,    419,
>   |  529,  625,  719,  729,  1024,  1619,   2048,   2187,   2401,   2809,
>   | 3671, 4096, 6561, 8192, 16384, 19683, 131072, 262144, 524288, 821641,
>   | 8388608,  33554432,  43046721,  134217728,  4294967296,  562949953421312,
>   | 9007199254740992,  18446744073709551616,  2417851639229258349412352,
>   | 2^128,  2^256,  2^512,  2^65536;
>   |
> --o------------------------------------------------------------------------------
> 
> 30 = p^1 p^2 p^3 = p.p< .p<
>                        p   p<
>                              p
> => |riff(30)| = 6
> 
> and 30 is the least number not in the above table,
> so we have Min Seq = 1, 2, 3, 5, 10, 15, 30, ...
> which is not in EIS.
> 
> Unless I have made the classic mistake of staying up too late again.
> 
> Which I will remedy, post haste.
> 
> Jon
> 
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤

I was holding back these sequences, but Jon has forced my hand...

I proffer this as the "core sequence" for riffs.

%I A000000
%S A000000
0,1,2,2,3,3,3,3,3,4,4,4,4,4,5,3,4,4,4,5,5,5,4,5,4,5,4,5,5,6,5,4,6,5,6,5,
%T A000000
5,5,6,6,5,6,5,6,6,5,6,5,4,5,6,6,4,5,7,6,6,6,5,7,5,6,6,4,7,7,5,6,6,7,6,6,
%U A000000
6,6,6,6,7,7,6,6,4,6,5,7,7,6,7,7,6,7,7,6,7,7,7,6,5,5,7,6,6,7,5,7,8,5,6,6
%N A000000 Nodes in riff (rooted index-functional forest) for n.
%F A000000 a(PROD(p_i^e_i)) = SUM(a(i)+a(e_i)+1), product over nonzero e_i in
prime factorizati
on.
%H A000000 J. Awbrey, <a
href="http://www.research.att.com/~njas/sequences/a061396a.txt">Illust
rations of riffs for small integers</a>
%C A000000 A061396(n) gives number of times n appears in this sequence.
%K A000000 nonn,easy
%O A000000 1,3
%A A000000 dww

Here is Jon's sequence extended somewhat.  I am still looking to compute this
more quickly.

%I A000001
%S A000001 1,2,3,5,10,15,30,55,105,165,330,660,1155,2145,4290,7755,15015,30030,
%T A000001 54285,100815,201630,403260,705705
%N A000001 Smallest k with n nodes in its riff (rooted index-functional forest)
for n.
%K A000001 nonn
%O A000001 0,2
%C A000001 Smallest k with A000000(k) = n.
%A A000001 Jon Aubrey (jawbrey at oakland.edu), dww