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Let m be an odd positive number. Let S_m denote the sequence {Product_{i=1..r} q_(n+t_i)}A050376%20and%20Sum">{n>=1}, where {q_i} is sequence A050376 and Sum{i=1..r} 2^(t_1 - t_i) is the binary representation of m, such that t_1 > t_2 > ... > t_r = 0. Note that {S_1, S_3, S_5, ...} is a partition of all integers > 1. Then S_1=A050376, which is obtained when we set r=1, t_1 = 0. [Formula made compatible with A240535 data by Peter Munn, Aug 10 2021]

This present sequence is S_3 in this partition. It is obtained when we set r=2, t_1=1, t_2=0.

S_m(n) = A052330(A030101(m)*2^(n-1)) = A329330(A050376(n), A052330(A030101(m))). - Peter Munn, Aug 10 2021

A minimal set of generators for A000379 as a group under A059897(.,.). - Peter Munn, Aug 11 2019

See comments in A240521.

See case r=2, t_1=2, t_2=0 in comment in A240521.

See case r=3, t_1=2, t_2=1, t_3=0 of comment in A240521.

See case r=2, t_1=3, t_2=0 in comment in A240521.

See case r=3, t_1=3, t_2=1, t_3=0 in comment in A240521. [Subscripts made consistent by Peter Munn, Oct 11 2021]

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.