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

A059897-based definition: (Start)

(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(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(n,7) = A(7,n) = A300841^4(n).

A(n,9) = A(9,n) = A300841^5(n).

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

KEYWORD

nonn,tabl

AUTHOR

Peter Munn, Nov 10 2019

STATUS

approved