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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. 29
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
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 squarefree 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 first power, thus sending 8 = 1:3 to 2 = 1:1, 27 = 2:3 to 3 = 2:1 and so on.
LINKS
J. Awbrey, Riffs and Rotes
FORMULA
If k = Product p_i^e_i, A(n,k) = p_i^A286561(n, A000040(e_i)), where A286561(x,y) gives the y-valuation of x. - Antti Karttunen, Nov 16 2019
EXAMPLE
` ` ` ` ` ` ` ` ` ` `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
Primal codes of finite partial functions on positive integers:
1 = { }
2 = 1:1
3 = 2:1
4 = 1:2
5 = 3:1
6 = 1:1 2:1
7 = 4:1
8 = 1:3
9 = 2:2
10 = 1:1 3:1
11 = 5:1
12 = 1:2 2:1
13 = 6:1
14 = 1:1 4:1
15 = 2:1 3:1
16 = 1:4
17 = 7:1
18 = 1:1 2:2
19 = 8:1
20 = 1:2 3:1
From Antti Karttunen, Nov 16 2019: (Start)
When the sequence is viewed as a square array read by falling antidiagonals, the top left 15 X 15 corner looks like this:
k= | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
----+--------------------------------------------------------------------
n= 1| 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
2| 1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15,
3| 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1,
4| 1, 4, 9, 1, 25, 36, 49, 1, 1, 100, 121, 9, 169, 196, 225,
5| 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1,
6| 1, 2, 3, 2, 5, 6, 7, 1, 3, 10, 11, 6, 13, 14, 15,
7| 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
8| 1, 8, 27, 1, 125, 216, 343, 1, 1, 1000, 1331, 27, 2197, 2744, 3375,
9| 1, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 4, 1, 1, 1,
10| 1, 2, 3, 1, 5, 6, 7, 2, 1, 10, 11, 3, 13, 14, 15,
11| 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
12| 1, 4, 9, 2, 25, 36, 49, 1, 3, 100, 121, 18, 169, 196, 225,
13| 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
14| 1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15,
15| 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1,
(End)
PROG
(PARI)
up_to = 105;
A106177sq(n, k) = { my(f = factor(k)); prod(i=1, #f~, f[i, 1]^valuation(n, prime(f[i, 2]))); };
A106177list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A106177sq(col, (a-(col-1))))); (v); };
v106177 = A106177list(up_to);
A106177(n) = v106177[n]; \\ Antti Karttunen, Nov 16 2019
CROSSREFS
Rows 1 - 3: A000012, A055231, A329376.
Columns 1 - 2: A000012, A006519.
Sequence in context: A248156 A352780 A331368 * A211985 A211009 A337266
KEYWORD
nonn,tabl,look
AUTHOR
Jon Awbrey, May 23 2005
STATUS
approved

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)