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A329330
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Multiplication operation of a ring over the positive integers that has A059897(.,.) as addition operation and is isomorphic to GF(2)[x] with polynomial x^i mapped to A050376(i+1). Square array read by descending antidiagonals: A(n,k), n >= 1, k >= 1.
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1
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 4, 1, 1, 5, 5, 5, 5, 1, 1, 6, 7, 7, 7, 6, 1, 1, 7, 12, 9, 9, 12, 7, 1, 1, 8, 9, 20, 11, 20, 9, 8, 1, 1, 9, 15, 11, 35, 35, 11, 15, 9, 1, 1, 10, 11, 28, 13, 8, 13, 28, 11, 10, 1, 1, 11, 21, 13, 45, 63, 63, 45, 13, 21, 11, 1
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OFFSET
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1,5
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COMMENTS
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When creating A329329, the author realized it was isomorphic to multiplication in the GF(2)[x,y] polynomial ring. However, A329329 was unusual in having A059897(.,.) as additive operator, whereas the equivalent univariate polynomial ring, GF(2)[x], is more commonly mapped (to integers) with bitwise exclusive-or (A003987) representing polynomial addition (and A048720(.,.) representing polynomial multiplication). This sequence shows how multiplication in GF(2)[x] can look when mapped to integers with A059897(.,.) representing polynomial addition.
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 the polynomial ring GF(2)[x]. 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. The most meaningful generating set for the additive group of GF(2)[x] is {x^i: i >= 0).
Using f to denote the intended isomorphism from GF(2)[x], we specify f(x^i) = A050376(i+1). This maps minimal generating sets of the additive groups, so the definition of f is completed by specifying f(a+b) = A059897(f(a), f(b)). We then calculate the image under f of polynomial multiplication in GF(2)[x], giving us this sequence as the matching multiplication operation for an isomorphic ring over the positive integers. Using g to denote the inverse of f, A(n,k) = f(g(n) * g(k)).
Note that A050376 is closed with respect to A(.,.).
Recall that GF(2)[x] is more usually mapped to integers with A003987(.,.) as addition and A048720(.,.) as multiplication. With this usual mapping, under which A000079(i) is the image of x^i, A052330(.) is the relevant isomorphism from nonnegative integers under A048720(.,.) and A003987(.,.) to positive integers under A(.,.) and A059897(.,.), with A052331(.) its inverse.
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LINKS
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Eric Weisstein's World of Mathematics, Group
Eric Weisstein's World of Mathematics, Ring
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FORMULA
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(End)
Derived identities: (Start)
A(n,1) = A(1,n) = 1.
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 4 5 7 12 9 15 11 21 13 20
4 | 1 4 5 7 9 20 11 28 13 36 16 35
5 | 1 5 7 9 11 35 13 45 16 55 17 63
6 | 1 6 12 20 35 8 63 120 99 210 143 15
7 | 1 7 9 11 13 63 16 77 17 91 19 99
8 | 1 8 15 28 45 120 77 14 117 360 176 420
9 | 1 9 11 13 16 99 17 117 19 144 23 143
10 | 1 10 21 36 55 210 91 360 144 22 187 756
11 | 1 11 13 16 17 143 19 176 23 187 25 208
12 | 1 12 20 35 63 15 99 420 143 756 208 28
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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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