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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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Table of n, a(n) for n=1..78.
Eric Weisstein's World of Mathematics, Distributive
Eric Weisstein's World of Mathematics, Group
Eric Weisstein's World of Mathematics, Ring
Wikipedia, Generating set of a group
Wikipedia, Polynomial ring
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FORMULA
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A(n,k) = A052330(A048720(A052331(n), A052331(k))), n >= 1, k >= 1.
A059897-based definition: (Start)
A(A050376(i), A050376(j)) = A050376(i+j-1).
A(A059897(n,k), m) = A059897(A(n,m), A(k,m)).
A(m, A059897(n,k)) = A059897(A(m,n), A(m,k)).
(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)
A(A300841(n), k) = A(n, A300841(k)) = A300841(A(n,k)).
A(n,3) = A(3,n) = A300841(n).
A(n,4) = A(4,n) = A300841^2(n).
A(n,5) = A(5,n) = A300841^3(n).
A(A050376(m), 6) = A(6, A050376(m)) = A240521(m).
A(n,7) = A(7,n) = A300841^4(n).
A(A050376(m), 8) = A(8, A050376(m)) = A240522(m).
A(n,9) = A(9,n) = A300841^5(n).
A(A050376(m), 10) = A(10, A050376(m)) = A240536(m).
A(A050376(m), 12) = A(12, A050376(m)) = A300841(A240521(m)).
A(A050376(m), 24) = A(24, A050376(m)) = A240524(m).
A(A050376(m), 30) = A(30, A050376(m)) = A241025(m).
A(A050376(m), 40) = A(40, A050376(m)) = A241024(m).
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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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Cf. A000079, A050376, A329329.
Distributes over A059897, and isomorphic to A048720 over A003987, with A052331 (inverse A052330) as isomorphism.
Row/column 3: A300841.
Row/column k sorted into increasing order: A003159 (k=3), A339690 (k=4), A000379 (k=6).
Subsequences of row/column k: A240521 (k=6), A240522 (k=8), A240536 (k=10), A240524 (k=24), A241025 (k=30), A241024 (k=40).
Sequence in context: A178059 A116188 A318274 * A049695 A096589 A176427
Adjacent sequences: A329327 A329328 A329329 * A329331 A329332 A329333
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KEYWORD
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nonn,tabl
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AUTHOR
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Peter Munn, Nov 10 2019
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STATUS
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approved
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