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 A329330 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. 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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. LINKS 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 FORMULA 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). EXAMPLE 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 CROSSREFS 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 KEYWORD nonn,tabl AUTHOR Peter Munn, Nov 10 2019 STATUS approved

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Last modified August 17 00:01 EDT 2022. Contains 356180 sequences. (Running on oeis4.)