Riffs & Rotes
Jon Awbrey
jawbrey at att.net
Sun May 29 19:16:31 CEST 2005
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R&R. Note 7
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| Numph, in thy orisons, be all my sequences renumbered ...
In order to make way for some long blocks of meta-sequences,
the (n o m)-composition tables had to be re-indexed like so:
A106177 = primal code composition table
http://www.research.att.com/projects/OEIS?Anum=A106177
Functional composition table for "n o m" = "n composed with m",
where n and m are the "primal codes" of finite partial functions
on the positive integers and 1 is the code for the empty function.
The right diagonal labeled by the prime power of the form j:k = (prime(j))^k
contains the j^th power primes in the factorization raised to the k^th power.
For example, the right diagonal labeled by the number 2 = 1:1 = (prime(1))^1
contains the power-free parts of each positive integer, specifically A055231,
and the right diagonal labeled by the number 4 = 1:2 = (prime(1))^2 contains
the squares of the square-free parts of positive integers. In general, then,
the right diagonal labeled by m = (j_i : k_i)_i = Product_i prime(j_i)^(k_i)
contains the product over i of the (j_i)th power primes in the factorization
raised to the (k_i)th powers.
For example, the operator 5 = 3:1 extracts the 3rd power primes
in the factorization of each n and raises them to the 1st power,
thus sending 8 = 1:3 to 2 = 1:1, 27 = 2:3 to 3 = 2:1, and so on.
1
1 1
1 2 1
1 3 1 1
1 1 1 4 1
1 5 2 9 1 1
1 6 1 1 1 2 1
1 7 1 25 1 3 1 1
1 1 1 36 1 2 1 8 1
1 1 1 49 1 5 1 27 1 1
1 10 3 1 1 6 1 1 1 2 1
1 11 1 1 2 7 1 125 4 3 1 1
1 3 1 100 1 1 1 216 1 1 1 4 1
1 13 2 121 1 3 1 343 1 5 1 9 1 1
1 14 1 9 1 10 1 1 1 6 1 2 1 2 1
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` n o m
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 . 1
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 2 . 1 . 2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 3 . 1 . 1 . 3
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 . 1 . 2 . 1 . 4
` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` 5 . 1 . 3 . 1 . 1 . 5
` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` 6 . 1 . 1 . 1 . 4 . 1 . 6
` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` 7 . 1 . 5 . 2 . 9 . 1 . 1 . 7
` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8
` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` 9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9
` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` `10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10
` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` `11 . 1 . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 . 1 . 11
` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` `12 . 1 . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 12
` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` `13 . 1 . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 . 1 . 13
` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` `14 . 1 . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 . 1 . 14
` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` `15 . 1 . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 . 1 . 15
` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` `16 . 1 . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 . 1 . 16
A106178 = primal code composition table, minus margins
http://www.research.att.com/projects/OEIS?Anum=A106178
Functional composition table for "n o m" = "n composed with m",
where n and m are the "primal codes" of finite partial functions
on the positive integers and 1 is the code for the empty function,
but omitting the trivial values of 1 at the margins of the table.
This sequence is derived from A107532
by removing the "obvious" values of 1
at the margins of the triangular array.
2
3 1
1 1 4
5 2 9 1
6 1 1 1 2
7 1 25 1 3 1
1 1 36 1 2 1 8
1 1 49 1 5 1 27 1
10 3 1 1 6 1 1 1 2
11 1 1 2 7 1 125 4 3 1
3 1 100 1 1 1 216 1 1 1 4
13 2 121 1 3 1 343 1 5 1 9 1
14 1 9 1 10 1 1 1 6 1 2 1 2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` n o m
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 . 1
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 2 . ` . 2
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 3 . ` . ` . 3
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` ` 4 . ` . 2 . ` . 4
` ` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` ` 5 . ` . 3 . 1 . ` . 5
` ` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` ` 6 . ` . 1 . 1 . 4 . ` . 6
` ` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` ` 7 . ` . 5 . 2 . 9 . 1 . ` . 7
` ` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` ` 8 . ` . 6 . 1 . 1 . 1 . 2 . ` . 8
` ` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` ` ` 9 . ` . 7 . 1 . 25. 1 . 3 . 1 . ` . 9
` ` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` ` `10 . ` . 1 . 1 . 36. 1 . 2 . 1 . 8 . ` . 10
` ` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` ` `11 . ` . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 . ` . 11
` ` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` ` `12 . ` . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . ` . 12
` ` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` ` `13 . ` . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 . ` . 13
` ` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` ` `14 . ` . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 . ` . 14
` ` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` ` `15 . ` . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 . ` . 15
` ` `\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
` `16 . ` . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 . ` . 16
| N.J.A. Sloane,
|'On-Line Encyclopedia of Integer Sequences',
| In: http://www.research.att.com/~njas/sequences/
| Cf. http://www.research.att.com/projects/OEIS?Anum=A061396
| Cf. http://www.research.att.com/projects/OEIS?Anum=A062504
| Cf. http://www.research.att.com/projects/OEIS?Anum=A062537
| Cf. http://www.research.att.com/projects/OEIS?Anum=A062860
| Cf. http://www.research.att.com/projects/OEIS?Anum=A106177
| Cf. http://www.research.att.com/projects/OEIS?Anum=A106178
Jon Awbrey
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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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