[seqfan] Re: conjectured list of divisibility sequences

peter.luschny peter.luschny at googlemail.com
Mon Jan 10 12:05:52 CET 2011

> Richard Mathar wrote

> The millenium task
Ars longa, vita brevis,
> by pure heuristics testing
experimentum periculosum,
> is to figure out for each case whether the
> conjectured divisibility property is correct.
iudicium difficile. Occasio praeceps, so let us start.

If a sequence has the divisibility property it gives
immediately rise to another one via the rectangular
array of numbers a(n*i)/a(i).

The prototype in the database for this is A000045 giving
raise to A028412. I prefer to see a(n) in the first line
not in the first column, though.

I see that A000045 is not in your list and it is also
not in the OEIS index of divisibility sequences. Do I
misunderstand your definition?

Thanks for pointing to A181858. Except for n = 9 the
prime factors of A181858(n) are the primes <= floor((n-1)/2).
Using this fact the divisibility property can be proven.

And the rectangular array arising from A181858 starts

1,   1,       1,           1,                4,
1,   1,       4,          18,              576,
1,   4,      64,       43200,          5080320,
1,  18,   43200,    12700800,     131681894400,
1, 144, 1270080, 32920473600, 2294425328025600,

Cheers, Peter

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