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A329329
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Multiplicative operator of a ring over the positive integers that has A059897(.,.) as additive operator and is isomorphic to GF(2)[x,y] with A329050(i,j) the image of x^i * y^j.
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17
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 7, 16, 7, 6, 1, 1, 7, 15, 25, 25, 15, 7, 1, 1, 8, 11, 36, 11, 36, 11, 8, 1, 1, 9, 27, 49, 35, 35, 49, 27, 9, 1, 1, 10, 25, 64, 13, 10, 13, 64, 25, 10, 1, 1, 11, 21, 81, 125, 77, 77, 125, 81
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OFFSET
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1,5
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COMMENTS
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Square array A(n,k), n >= 1, k >= 1, read by descending antidiagonals.
The group defined by the binary operation A059897(.,.) over the positive integers is commutative with all elements self-inverse, and isomorphic to the additive group of GF(2) polynomial rings such as GF(2)[x,y]. There is a unique isomorphism extending each bijective mapping between respective minimal generating sets. The lexicographically earliest minimal generating set for the A059897 group is A050376, often called the Fermi-Dirac primes. This set has a natural arrangement in a square array, given by A329050(i,j) = prime(i+1)^(2^j), i >= 0, j >= 0. The most meaningful generating set for the additive group of GF(2)[x,y] is {x^i * y^j: i >= 0, j >= 0), which similarly forms a square array. All this makes A329050(i,j) especially appropriate to be the image (under an isomorphism) of the GF(2) polynomial x^i * y^j.
Using g to denote the intended isomorphism, we specify g(x^i * y^j) = A329050(i,j). This maps minimal generating sets of the additive groups, so the definition of g is completed by specifying g(a+b) = A059897(g(a), g(b)). We then calculate the image under g of polynomial multiplication in GF(2)[x,y], giving us this sequence as the matching multiplicative operator for an isomorphic ring over the positive integers. Using f to denote the inverse of g, A[n,k] = g(f(n) * f(k)).
See the formula section for an alternative definition based on the A329050 array, independent of GF(2)[x,y].
Closely related to A306697 and A297845. If A059897 is replaced in the alternative definition by A059896 (and the definition is supplemented by the derived identity for the absorbing element, shown in the formula section), we get A306697; if A059897 is similarly replaced by A003991 (integer multiplication), we get A297845. This sequence and A306697, considered as multiplicative operators, are carryless arithmetic equivalents of A297845. A306697 uses a method analogous to binary-OR when there would be a multiplicative carry, while this sequence uses a method analogous to binary exclusive-OR. In consequence A(n,k) <> A297845(n,k) exactly when A306697(n,k) <> A297845(n,k). This relationship is not symmetric between the 3 sequences: there are n and k such that A(n,k) = A306697(n,k) <> A297845(n,k). For example A(54,72) = A306697(54,72) = 273375000 <> A297845(54,72) = 22143375000.
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LINKS
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Eric Weisstein's World of Mathematics, Ring
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FORMULA
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Alternative definition: (Start)
(End)
Derived identities: (Start)
A(n,1) = A(1,n) = 1 (1 is an absorbing element).
A(n,2) = A(2,n) = n.
A(n,k) = A(k,n).
A(n, A(m,k)) = A(A(n,m), k).
(End)
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EXAMPLE
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Square array A(n, k) begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12
---+-------------------------------------------------------------
1| 1 1 1 1 1 1 1 1 1 1 1 1
2| 1 2 3 4 5 6 7 8 9 10 11 12
3| 1 3 5 9 7 15 11 27 25 21 13 45
4| 1 4 9 16 25 36 49 64 81 100 121 144
5| 1 5 7 25 11 35 13 125 49 55 17 175
6| 1 6 15 36 35 10 77 216 225 210 143 540
7| 1 7 11 49 13 77 17 343 121 91 19 539
8| 1 8 27 64 125 216 343 32 729 1000 1331 1728
9| 1 9 25 81 49 225 121 729 625 441 169 2025
10| 1 10 21 100 55 210 91 1000 441 22 187 2100
11| 1 11 13 121 17 143 19 1331 169 187 23 1573
12| 1 12 45 144 175 540 539 1728 2025 2100 1573 80
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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