# [seqfan] Re: A125121, A005360.

David Wilson davidwwilson at comcast.net
Mon Nov 18 02:32:36 CET 2013

```I think perhaps the best base generalization might be

n is base-b sturdy if the base-b digit sum of n = smallest base-b
digit sum of kn for k >= 1.

For example, the base-2 (standard) sturdy numbers are those for which
A000120(n) = A086342(n).
Likewise, the base-10 sturdy numbers would be those for which A007953(n) =
A077196(n).

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Charles
> Greathouse
> Sent: Sunday, November 17, 2013 11:26 AM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: A125121, A005360.
>
> 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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```