[seqfan] Re: A125121, A005360.

[Charles Greathouse]

That would be one generalization. Here's another:

A number n is q-sturdy if there is no multiple kn with fewer nonzero digits
in base q.

and another:

A number n is q-sturdy if there is no multiple kn with fewer 1 digits in
base q, fewer 2 digits in base q, ..., and fewer q-1 digits in base q.

and another:

A number n is q-sturdy if there is no multiple kn with fewer 1 digits in
base 1, fewer 1 and 2 digits in base q, ..., and fewer 1, 2, ..., q-1
digits in base q.

What makes the Hamming weight special (IMO) is that it generalizes these
and many more.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Sat, Nov 16, 2013 at 6:04 PM, Allan Wechsler <acwacw at gmail.com> wrote:

> I had a slight feeling that A125121 was basey, and was going to suggest a
> base-k analog:
>
> A number n is "sturdy in base q" if there is no multiple kn, the sum of
> whose digits in base q falls below the sum of the digits of n in base q.
>
> The sequence base-3 sturdy numbers begins 1, 2, 3, 4, 6 ...
>
> 7 is not sturdy in base 3; I haven't figured out 8 but I suspect it isn't
> sturdy either.
>
>
>
> On Sat, Nov 16, 2013 at 4:03 PM, Charles Greathouse <
> charles.greathouse at case.edu> wrote:
>
> > It's not clear whether keyword:base applies to both sequences or to
> > neither. (Certainly I agree that the two should be the same in that
> > respect.) The Hamming weight itself does not have keyword:base and I
> agree
> > with that here -- although it can be interpreted in a base-specific
> fashion
> > it need not be and it does seem more natural than most base sequences.
> > That, in turn, would suggest that perhaps neither of these sequences need
> > the keyword. But it's not entirely clear.
> >
> > Charles Greathouse
> > Analyst/Programmer
> > Case Western Reserve University
> >
> >
> > On Sat, Nov 16, 2013 at 1:04 AM, L. Edson Jeffery <lejeffery2 at gmail.com
> > >wrote:
> >
> > > Since there is ongoing discussion regarding these two sequences, should
> > > A125121 have the keyword "base" since it is the complement of A005360
> > which
> > > was given that keyword?
> > >
> > > Ed Jeffery
> > >
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