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A329329 Multiplicative operator of a ring over the positive integers that has A059897(.,.) as additive operator and is isomorphic to GF(2)[x,y] with A329050(i,j) the image of x^i * y^j.

%I A329329 #25 Feb 16 2025 08:33:58
%S A329329 1,1,1,1,2,1,1,3,3,1,1,4,5,4,1,1,5,9,9,5,1,1,6,7,16,7,6,1,1,7,15,25,
%T A329329 25,15,7,1,1,8,11,36,11,36,11,8,1,1,9,27,49,35,35,49,27,9,1,1,10,25,
%U A329329 64,13,10,13,64,25,10,1,1,11,21,81,125,77,77,125,81
%N A329329 Multiplicative operator of a ring over the positive integers that has A059897(.,.) as additive operator and is isomorphic to GF(2)[x,y] with A329050(i,j) the image of x^i * y^j.
%C A329329 Square array A(n,k), n >= 1, k >= 1, read by descending antidiagonals.
%C A329329 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 GF(2) polynomial rings such as GF(2)[x,y]. 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. This set has a natural arrangement in a square array, given by A329050(i,j) = prime(i+1)^(2^j), i >= 0, j >= 0. The most meaningful generating set for the additive group of GF(2)[x,y] is {x^i * y^j: i >= 0, j >= 0}, which similarly forms a square array. All this makes A329050(i,j) especially appropriate to be the image (under an isomorphism) of the GF(2) polynomial x^i * y^j.
%C A329329 Using g to denote the intended isomorphism, we specify g(x^i * y^j) = A329050(i,j). This maps minimal generating sets of the additive groups, so the definition of g is completed by specifying g(a+b) = A059897(g(a), g(b)). We then calculate the image under g of polynomial multiplication in GF(2)[x,y], giving us this sequence as the matching multiplicative operator for an isomorphic ring over the positive integers. Using f to denote the inverse of g, A[n,k] = g(f(n) * f(k)).
%C A329329 See the formula section for an alternative definition based on the A329050 array, independent of GF(2)[x,y].
%C A329329 Closely related to A306697 and A297845. If A059897 is replaced in the alternative definition by A059896 (and the definition is supplemented by the derived identity for the absorbing element, shown in the formula section), we get A306697; if A059897 is similarly replaced by A003991 (integer multiplication), we get A297845. This sequence and A306697, considered as multiplicative operators, are carryless arithmetic equivalents of A297845. A306697 uses a method analogous to binary-OR when there would be a multiplicative carry, while this sequence uses a method analogous to binary exclusive-OR. In consequence A(n,k) <> A297845(n,k) exactly when A306697(n,k) <> A297845(n,k). This relationship is not symmetric between the 3 sequences: there are n and k such that A(n,k) = A306697(n,k) <> A297845(n,k). For example A(54,72) = A306697(54,72) = 273375000 <> A297845(54,72) = 22143375000.
%H A329329 Rémy Sigrist, <a href="/A329329/a329329.gp.txt">PARI program for A329329</a>
%H A329329 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Ring.html">Ring</a>
%H A329329 Wikipedia, <a href="https://en.wikipedia.org/wiki/Generating_set_of_a_group">Generating set of a group</a>
%H A329329 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polynomial_ring">Polynomial ring</a>
%F A329329 Alternative definition: (Start)
%F A329329 A(A329050(i_1, j_1), A329050(i_2, j_2)) = A329050(i_1+i_2, j_1+j_2).
%F A329329 A(A059897(n,k), m) = A059897(A(n,m), A(k,m)).
%F A329329 A(m, A059897(n,k)) = A059897(A(m,n), A(m,k)).
%F A329329 (End)
%F A329329 Derived identities: (Start)
%F A329329 A(n,1) = A(1,n) = 1 (1 is an absorbing element).
%F A329329 A(n,2) = A(2,n) = n.
%F A329329 A(n,k) = A(k,n).
%F A329329 A(n, A(m,k)) = A(A(n,m), k).
%F A329329 (End)
%F A329329 A(A019565(i), 2^j) = A019565(i)^j = A329332(i,j).
%F A329329 A(A225546(i), A225546(j)) = A225546(A(i,j)).
%F A329329 A(n,k) = A306697(n,k) = A297845(n,k), for n = A050376(i), k = A050376(j).
%F A329329 A(n,k) <= A306697(n,k) <= A297845(n,k).
%F A329329 A(n,k) < A297845(n,k) if and only if A306697(n,k) < A297845(n,k).
%e A329329 Square array A(n, k) begins:
%e A329329   n\k|  1   2   3    4    5    6    7     8     9    10    11    12
%e A329329   ---+-------------------------------------------------------------
%e A329329     1|  1   1   1    1    1    1    1     1     1     1     1     1
%e A329329     2|  1   2   3    4    5    6    7     8     9    10    11    12
%e A329329     3|  1   3   5    9    7   15   11    27    25    21    13    45
%e A329329     4|  1   4   9   16   25   36   49    64    81   100   121   144
%e A329329     5|  1   5   7   25   11   35   13   125    49    55    17   175
%e A329329     6|  1   6  15   36   35   10   77   216   225   210   143   540
%e A329329     7|  1   7  11   49   13   77   17   343   121    91    19   539
%e A329329     8|  1   8  27   64  125  216  343    32   729  1000  1331  1728
%e A329329     9|  1   9  25   81   49  225  121   729   625   441   169  2025
%e A329329    10|  1  10  21  100   55  210   91  1000   441    22   187  2100
%e A329329    11|  1  11  13  121   17  143   19  1331   169   187    23  1573
%e A329329    12|  1  12  45  144  175  540  539  1728  2025  2100  1573    80
%o A329329 (PARI) \\ See Links section.
%Y A329329 Cf. A050376, A019565, A329332.
%Y A329329 A059897, A225546, A329050 are used to express relationship between terms of this sequence.
%Y A329329 Related binary operations: A297845/A003991, A306697/A059896.
%K A329329 nonn,tabl
%O A329329 1,5
%A A329329 _Peter Munn_, Nov 11 2019