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, 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

Offset: 1

Author

Peter Munn, Nov 10 2019

Keywords

• nonn
• tabl

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.

Examples

			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
		

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

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).

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).