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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 Cf. A000040, A061396, A062504, A062537, A062860, A106178, A181819, A286561. Rows 1 - 3: A000012, A055231, A329376. Columns 1 - 2: A000012, A006519. Sequence in context: A202241 A248156 A331368 * A211985 A211009 A337266 Adjacent sequences:  A106174 A106175 A106176 * A106178 A106179 A106180 KEYWORD nonn,tabl,look AUTHOR Jon Awbrey, May 23 2005 STATUS approved

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Last modified July 24 14:51 EDT 2021. Contains 346273 sequences. (Running on oeis4.)