185 is special

[Jonathan Post]

185 is the number of conjugacy classes in the automorphism group of
the 8 dimensional hypercube.

So it is the first value in the sequence: number of conjugacy classes
in the automorphism group of the 8^n dimensional hypercube.

There are sneaky ways to make a sequence with an apparently different
definition which is provably equal to that.



Well, I'll stand up for lowly k-gonal numbers as being of interest (to me,
anyways). Generally, there are two questions I'd like to answer. Someone may
have already answered them, if so I haven't found those answers yet:


1.) If you know the factors of some composite integer k, can you find all
the representations of k as a sum of constant-spaced integers? For instance,
35 = 5*7, has at least the following representations:
{7 7 7 7 7; 5 6 7 8 9; 3 5 7 9 11; 1 4 7 10 13; 5 5 5 5 5 5 5; 2 3 4 5 6 7
8}. Is that all of them? Is there an algorithmic way to determine all of
them? I believe the answer to that last question is yes, but I haven't
formulated a complete method yet. I have a vague inkling that the "ease" of
factoring k may have something to do with the ratio of the number of these
represenations of k to k, but it's just an idea at this point.

2.) Is the converse true; if you know one of the representations of k as the
time) determine the others? I'm pretty sure it can be done generally (I've
found a few ways that work for different types of numbers), I'm just not


In the case of Zak's sequence it's a subset of A117048 -- Prime numbers
which are expressible as the sum of two triangular numbers -- since 10 is a
triangular number. With respect to the sequence proposed by Jonathan, it
interests me but I agree that there ought to be a way to demonstrate why
it's interesting, aside from it being a companion sequence to A125585 (Array
of constant-spaced integers read by antidiagonals). For me I'm generally
trying to amass information about sums of constant-spaced integers, but I
can certainly understand why that might not interest most people.



-----Original Message-----

Dear Zak, I don't mind the sequence. I very much respect Neil's
keywording, even when he gives me a "less" or "dumb."

However, your "10" is rather arbitrary.  Since 10 is itself a
triangular number, I've submitted the slightly more general:

NEW SEQUENCE FROM Jonathan Vos Post


%I A000001
%S A000001 2, 3, 4, 1, 1, 7, 4, 1, 1, 7, 3
%N A000001 Least nonnegative m such that T(n) + T(m) is prime, for T(n)
= n(n+1)/2.
%C A000001 What is the simplest proof that this is defined for all
nonzero n?
%F A000001 a(n) = Min{m: m*(m+1)/2 + n*(n+1)/2 is prime}.
a(n) = Min{m: A000217(m) + A000217(n) is an element of A000040}.
%e A000001 a(6) = 4 because T(4) = 10 is the least triangular number
whose sum with T(6) = 21 is prime, since {21+0 = 3*7, 21+3 = 2^3*3, 21+6
= 3^3} are all composite, but 21+10 = 31 is prime.
%Y A000001 Cf. A000040, A000217, A129755, A130334.
%O A000001 0,1
%K A000001 ,easy,more,nonn,
%A A000001 Jonathan Vos Post (jvospost2 at yahoo.com), May 31 2007
RH
RA 192.20.225.32

Also, I feel your pain regarding email and password problems.  These
are curses of the late 20th and early 21st century.

An even more general sequence would be a table, by antidiagonals, of
the same but where "k-gonal number" replaces "triangular number." Yet
that does not inherently interest me, either, absent a nice proof or
asymptotic result.

Best,

Jonathan