Riffs & Rotes & A061396
David W. Wilson
wilson at aprisma.com
Mon Jun 25 21:44:39 CEST 2001
Jon Awbrey wrote:
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> David W. Wilson wrote:
>
> JA: Re:
>
> ...
>
> 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------------------------------------------------------------------------------
>
> JA: 30 = p^1 p^2 p^3 = p.p< .p<
> p p<
> p
>
> I meant, of course, to write:
>
> 30 = p_1 p_2 p_3 = p.p< .p<
> p p<
> p
>
> JA: => |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.
>
> DW: I was holding back these sequences, but Jon has forced my hand ...
>
> Sorry, David, I am new here, and had no clue that
> some brand of poker was being played. I was just
> treating it like all of my other discussion groups,
> where I am also much too chatty for anybody's good,
> but where nothing much is at stake but time, all too
> precious time. I think that I indicated in my initial
> note that this is no longer my active arena, and my whole
> reason for being here is turn this old hobby horse of mine
> over to abler cowhands, of whose ability I already have more
> than ample evidence. So, count the ways, and do as you will!
> As long as I can keep it recreational, I will skulk about the
> old corral, and kibbitz on what all's going on, but I already
> had my fill of this damned ole rodeo in my own wrangler days.
> But I would, for a little while, like to try and comprehend
> what is sprung from these seeds, so if you could explain
> what is meant by a "core sequence", I would like that.
> DW: Here is Jon's sequence extended somewhat.
> I am still looking to compute this more quickly.
>
> DW:
> | %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 Awbrey (jawbrey at oakland.edu), dww
Let R(n) be the number nodes in the riff of n. Then we have
R(n) = R(PROD(p_i^e_i)) = SUM(R(p_i^e_i)) = SUM(R(i) + R(e_i) + 1).
Now suppose we wish to find the smallest n with R(n) = k. Then we have
k = SUM(R(p_i^e_i))
that is, the R(p_i^e_i) form a partition of k. Thus, to find n, we must
run through the partitions {a_j} of k, choose all possible distinct i
and e_i with R(p_i^e_i) = a_j, and minimize n = PROD(p_i^e_i) over these
i and e_i.
In general, there are a large number of prime powers p_i^e_i with
R(p_i^e_i) = a_j. However, given that our aim is to minimize n, we
can neglect all but the smallest few p_i^e_i for each a_j. Doing
this makes the computation of n tractable, and allows us to extend the
above sequence to:
1 2 3 5 10 15 30 55 105 165 330 660 1155 2145 4290 7755 15015 30030
54285 100815 201630 403260 705705 1411410 2822820 5645640 11392095
20465445 40930890 79744665 159489330 318978660 637957320 1321483020
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