[seqfan] Re: Problem connected to A000960

[David Wilson]

Consider the family of sequences parameterized on integer k > 0:

a(0) = k.
a(n) = smallest multiple of n that is >= a(n-1) (n >= 1).

For example, starting with a(0) = k = 100, we obtain the sequence

a =
(100,100,100,102,104,105,108,112,112,117,120,121,132,143,154,165,176,187,198
,209,...)

Starting at 121, each successive element is 11 more than the previous. We
can prove this continues forever, that is, a(n) = 11n for all n >= 11.

For general starting value k, if m is the first element with a(m) <= m^2,
then the first difference is d = a(m)/m for subsequent elements, so that
a(n) = d*n for n >= m.

I was interested in d as a function of k, so I computed d(k) for a few small
values of k >= 1:

d = (1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,...)

d increases at the values

k =
(1,3,7,13,19,27,39,49,63,79,91,109,133,147,181,207,223,253,289,307,349,387,.
..)

which I am pretty sure is A000960, but I couldn't prove it.