%I #31 Aug 23 2021 13:39:16
%S 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,
%T 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,
%U 8,13,28,11,10,1,1,11,21,13,45,63,63,45,13,21,11,1
%N 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.
%C 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.
%C 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).
%C 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)).
%C Note that A050376 is closed with respect to A(.,.).
%C 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Distributive.html">Distributive</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Group.html">Group</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Ring.html">Ring</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Generating_set_of_a_group">Generating set of a group</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polynomial_ring">Polynomial ring</a>
%F A(n,k) = A052330(A048720(A052331(n), A052331(k))), n >= 1, k >= 1.
%F A059897-based definition: (Start)
%F A(A050376(i), A050376(j)) = A050376(i+j-1).
%F A(A059897(n,k), m) = A059897(A(n,m), A(k,m)).
%F A(m, A059897(n,k)) = A059897(A(m,n), A(m,k)).
%F (End)
%F Derived identities: (Start)
%F A(n,1) = A(1,n) = 1.
%F A(n,2) = A(2,n) = n.
%F A(n,k) = A(k,n).
%F A(n, A(m,k)) = A(A(n,m), k).
%F (End)
%F A(A300841(n), k) = A(n, A300841(k)) = A300841(A(n,k)).
%F A(n,3) = A(3,n) = A300841(n).
%F A(n,4) = A(4,n) = A300841^2(n).
%F A(n,5) = A(5,n) = A300841^3(n).
%F A(A050376(m), 6) = A(6, A050376(m)) = A240521(m).
%F A(n,7) = A(7,n) = A300841^4(n).
%F A(A050376(m), 8) = A(8, A050376(m)) = A240522(m).
%F A(n,9) = A(9,n) = A300841^5(n).
%F A(A050376(m), 10) = A(10, A050376(m)) = A240536(m).
%F A(A050376(m), 12) = A(12, A050376(m)) = A300841(A240521(m)).
%F A(A050376(m), 24) = A(24, A050376(m)) = A240524(m).
%F A(A050376(m), 30) = A(30, A050376(m)) = A241025(m).
%F A(A050376(m), 40) = A(40, A050376(m)) = A241024(m).
%e Square array A(n,k) begins:
%e n\k | 1 2 3 4 5 6 7 8 9 10 11 12
%e ----+----------------------------------------------------------
%e 1 | 1 1 1 1 1 1 1 1 1 1 1 1
%e 2 | 1 2 3 4 5 6 7 8 9 10 11 12
%e 3 | 1 3 4 5 7 12 9 15 11 21 13 20
%e 4 | 1 4 5 7 9 20 11 28 13 36 16 35
%e 5 | 1 5 7 9 11 35 13 45 16 55 17 63
%e 6 | 1 6 12 20 35 8 63 120 99 210 143 15
%e 7 | 1 7 9 11 13 63 16 77 17 91 19 99
%e 8 | 1 8 15 28 45 120 77 14 117 360 176 420
%e 9 | 1 9 11 13 16 99 17 117 19 144 23 143
%e 10 | 1 10 21 36 55 210 91 360 144 22 187 756
%e 11 | 1 11 13 16 17 143 19 176 23 187 25 208
%e 12 | 1 12 20 35 63 15 99 420 143 756 208 28
%Y Cf. A000079, A050376, A329329.
%Y Distributes over A059897, and isomorphic to A048720 over A003987, with A052331 (inverse A052330) as isomorphism.
%Y Row/column 3: A300841.
%Y Row/column k sorted into increasing order: A003159 (k=3), A339690 (k=4), A000379 (k=6).
%Y Subsequences of row/column k: A240521 (k=6), A240522 (k=8), A240536 (k=10), A240524 (k=24), A241025 (k=30), A241024 (k=40).
%K nonn,tabl
%O 1,5
%A _Peter Munn_, Nov 10 2019