[seqfan] Re: Convert/concatenate/divide

[jnthn stdhr]

On Apr 25, 2018 4:45 AM, "M. F. Hasler" <seqfan at hasler.fr> wrote:
>
> On Tue, Apr 24, 2018 at 3:18 PM, jnthn stdhr <jstdhr at gmail.com> wrote:
> > For n > 0, covert n to its binary representation b, then let x equal the
> > concatenation of the decimal values of each power of two <= n muliplied
by
> > its associated bit in b. If n divides x  a(n) = n.
>
> You mean, "add n to the list"  (according to:
> > (...) The sequence begins:  1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18,(...)
>

Yes.

> There are several variants:
>
> - omit the 0's : concatenate only the values corresponding to nonzero bits
>
> - instead of using the "decimal value of each power of two", use 1,2,3,...
>   (or rather ...3,2,1) (*)
>

Combining the two above might be worth adding, because even numbers are
curiously rare(ish).  Up to k=100000:  1, 5, 7, 15, 21, 219, 225, 281, 305,
332, 843, 1393, 16955, 64847, 65023, 65407, 65470, 65471, 65503, 65519,
65527, 65531, 65533, 65534, 65535, 68188, 81919, 97791, 98173, 98287,
98295, 98301, 98302, 98303,...

with x-values beginning:  0,20,210,3210,420,764310,...

> - or another "canonical" sequence, e.g., the primes
>
> - instead of concatenation, multiply or add (this would give back n
> for "powers of two")
>
> - consider the sequence(s)  x(n), not only { n :  n | x(n) }.
>   E.g., (*) would yield:
>   "concatenation of the indices of nonzero bits, starting with 1 for
> the most/ least significant bit."
>   (could be concatenated in either direction)
>
> Maybe not all of these (combinations) merit being added: these are
> just some ideas...
>
> - Maximilian

Good ideas!

>
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