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.

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

Offset: 1

Author

Jon Awbrey, May 23 2005

Keywords

• nonn
• tabl
• look

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.

Examples

			` ` ` ` ` ` ` ` ` ` `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)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..25425; the first 225 antidiagonals of the array
• J. Awbrey, Riffs and Rotes

Crossrefs

Cf. A000040, A061396, A062504, A062537, A062860, A106178, A181819, A286561.

Rows 1 - 3: A000012, A055231, A329376.

Columns 1 - 2: A000012, A006519.

Programs

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

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