[seqfan] Erdös-Woods numbers == prime-partitionable numbers

[Chris]

Dear Seqfans,

 

I believe I have the basis for a proof that there is a 1-to-1 correspondence
between the Erdös-Woods numbers and the prime-partitionable numbers.

 

Definitions from A059756:

An Erdös-Woods number is the length of an interval of consecutive integers
(including the smaller end-point) with the property that every element has a
factor in common with one of the end-points, where “length of an interval”
is the total number of terms in the interval including the end-points, minus
1.

A prime-partitionable  number n is such that there is a partition {P1, P2}
of the primes less than n such that for any composition n1+n2=n, there is
(p1,p2) in P1 x P2 such that p1|n1 or p2|n2.

 

If the length of a known Erdös-Woods interval is n, then it is necessary and
sufficient that one end-point has the primes in P1 as factors and the other
end-point has the primes in P2 as factors.

The n-1 compositions will exhibit the same prime factors in the same or the
reverse order as the n-1 consecutive integers in the Erdös-Woods interval
beyond the initial end-point, depending on whether the smaller end-point
contains the primes in P1 or P2 as factors.

 

In addition:

Definition from A130173:

A finite sequence of consecutive positive integers is called "stapled" if
each element in the sequence is not relatively prime to at least one other
element in the sequence.

 

A stapled interval includes both end-points.

Hence, an Erdös-Woods interval plus the larger end-point is a stapled
interval by definition.

Consequently, as  <http://oeis.org/wiki/User:Fidel_I._Schaposnik> Fidel I.
Schaposnik points out in A194585, the smaller end-points of Erdös-Woods
intervals with the same length can be calculated by recurrence once the
smaller end-points of the intervals less than #n have been determined.

 

Best regards,

Chris Gribble