[seqfan] Integers split into 2 classes, and a "dark horse" sequence

[Peter Munn]

dark horse: a usually little-known contender ... that makes an
unexpectedly good showing [Merriam-Webster]

Dear seqfans,

As one or two of you will be aware, I have been working on
cross-referencing sequences that can be defined as the lexicographically
earliest sequence of distinct nonnegative integers such that no term is
the result of a specified function applied to any terms of the sequence
(or, alternatively, any distinct terms).

Several resulting sequences come under the heading of 2-way classification
of integers, the defined sequence being in some sense "odd" and its
complement "even". The odd numbers themselves are the result when the
function is "addition of 2 terms"; whereas multiplication generates
A026424 (products of an odd number of prime factors); and binary exclusive
OR generates A000069 (numbers with an odd number of 1-bits).

But what of A000028, numbers that have an odd total of 1-bits in their
canonical factorisation's exponents? From A000028's description of
{A000028,A000379}, I infered it to be a well-known and studied 2-way
classification of the integers that I should try to encompass. Moreover,
A000028 can be defined as the products of odd numbers of distinct terms of
A050376, for which the designation "Fermi-Dirac primes" is gaining
popularity. In this sense, A000028 is an analogue of the aforementioned
A026424.

However, I had tried various combinations of standard functions with my
"lexicographically earliest" formulations, and got no closer than
generating a sequence that starts similarly but diverges, namely
A026416\{1}. Instead, a much less obvious function proved suitable:
A059897. But the more I looked at it, the more I asked myself "has my
search brought me to an uncut diamond?"

As a function, A059897 is defined in terms of binary exclusive OR applied
to the respective prime exponents in its operands' factorisations. This
doesn't bring out its relevance very well: imagine integer multiplication
described by the bit operations used to calculate it.

A059897 is actually the group operation on the positive integers
considered as an Abelian group generated by A050376, the "Fermi-Dirac
primes", with every non-identity element having order 2. It seems
reasonable to describe the group operation loosely as "multiplication" in
that it matches the result of standard integer multiplication when its
operands have no common A050376 factors.

I find the "multiplication" table of great interest -- its 12 X 12
top-left is in A059897's example section. Every row/column is, of course,
a permutation of the positive integers, and three seem to have been added
to OEIS quite independently (A073675, A120229, A120230).

It has appealing variety in that some of the entries match the result of
integer division, rather than multiplication, because any two non-identity
elements p and q will generate a Klein 4-group {1,p,q,pq}; and where the
A050376 factors of p and q overlap, their generated subgroup "swaps
around" these factors, for example the subgroup {1,6,10,15} with 6x10 =
15, 6x15 = 10 and 10x15 = 6.

I suspect A050376 is heading towards designation as an OEIS core sequence
before long, and my current fascination with A059897 tempts me to view it
as a possible "dark horse" to rise to similar status in the very long
term. For now, perhaps it deserves the keyword "nice"?

Best Regards,

Peter


P.S. Looking through those sequences designated "core", should one of the
integer multiplication table sequences have the keyword? There are several
tables with the designation, headed by Pascal's triangle.