The Human Metabolome Database

[N. J. A. Sloane]

Rick L. Shepherd's A127637 "Smallest square-free triangular number
with n prime factors" invites more than a one-shot comment.

It can be seen as the top row of an infinite array, whose kth row, for k>2, is
a(k,n) = Smallest square-free k-gonal number with n prime factors.

3 | 1, 3, 6, 66, 210, 3570, 207690, 930930, 56812470, 1803571770,
32395433070,...
4 | 0 [suggests tweak in definition]
5 | 0, 5, 22, 70, 210, 11310, ...
etc.

It is not coincidence that 210 is in both rows 3 and 5.

Formal definitions invoke the sequence of square-free integers
intersected with the k-gonalk numbers, and the omega function.

The array eliminates the slightly arbitrary singling-out of triangular numbers.



I'd like to chime in a bit here; I think the figurate numbers generally are
treated singly instead of considered as a coherent group. Oftentimes,
however, they can be used interchangably. For instance in Fermat's factoring
method we look for solutions to:

x^2 == y^2 mod n

This, of course, is also true for other figurate numbers, not just squares
(although square solutions may be the easiest to find). I recently submitted
this sequence -- A125585 "Array of constant-spaced integers read by
antidiagonals" -- which contains all of the constant-spaced integers, the
will spark a bit more interest in analyzing the set of figurate numbers as a
whole instead of in parts.

Which leads me to a question -- has anyone written a paper on how many ways
a particular integer can be expressed as the difference of two figurate
numbers? Of course all of the stuff related to squares and Fermat's method
would be included, but what about triangle numbers, pentagonal, hexagonal,
etc? It'd be nice to see a treatment that considered them all. I personally
believe that the number of factors of an integer and the closeness of those
factors has alot to do with how many representations there are. I suspect,
for instance, that the product of two large primes (where the two primes are
different by a bounded percentage) has one and only one representation as
the difference of squares and two representations as the difference of
triangle numbers. Fermat's method allows for multiples of n and that's where
the other representations (and therefore solutions) come into play. Anyways,
all of this may already be covered but I've yet to find any papers that
discuss it.




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

Rick L. Shepherd's A127637 "Smallest square-free triangular number
with n prime factors" invites more than a one-shot comment.

It can be seen as the top row of an infinite array, whose kth row, for k>2,
is
a(k,n) = Smallest square-free k-gonal number with n prime factors.

3 | 1, 3, 6, 66, 210, 3570, 207690, 930930, 56812470, 1803571770,
32395433070,...
4 | 0 [suggests tweak in definition]
5 | 0, 5, 22, 70, 210, 11310, ...
etc.

It is not coincidence that 210 is in both rows 3 and 5.

Formal definitions invoke the sequence of square-free integers
intersected with the k-gonalk numbers, and the omega function.

The array eliminates the slightly arbitrary singling-out of triangular
numbers.