[seqfan] Re: A325799 = "Signature excitation" of n?

[Olivier Gerard]

Dear David, Allan,

At this point I suggest you continue this discussion in private for a while
and then send to the list
a summary of your progress on the matter in a few days.

Olivier GERARD
seqfan list administrator


On Tue, Oct 11, 2022 at 12:00 PM Allan Wechsler <acwacw at gmail.com> wrote:

> David Corneth, your data looks right to me (now that my head has cleared a
> little). But I confess I can't understand your code ... or even know what
> language it's in.
>
> On Mon, Oct 10, 2022 at 10:38 PM David Corneth <davidacorneth at gmail.com>
> wrote:
>
> > As others said, I think A325799 is not it.
> > The sequence is has data 0, 0, 1, 0, 2, 0, 3, 0, 1, 1, 4, 0, 5, 2, 2, 0,
> 6,
> > 0, 7, 1, ... I think (and offset 1).
> > and a prog a(n) = if(n == 1, return(0)); my(f = factor(n)); sum(i = 1,
> #f~,
> > primepi(f[i, 1])) - binomial(#f~+1, 2)
> >
> > Or did I misunderstand?
> >
> > On Mon, Oct 10, 2022 at 3:23 PM Lucas J <lucas at spicyorange.com> wrote:
> >
> > > A325799 is not exactly the signature excitation of n. The excitation of
> > > 20 is 1, but A325799(20) is 0.
> > > On 2022-10-09 14:40, Allan Wechsler wrote:
> > > > If you write n as p1^e1 * p2^e2 * ... * pk^ek, where p1, p2, p3, ...
> > > > are
> > > > the primes which divide n (in order of size), then by "prime
> signature"
> > > > I
> > > > mean the k-tuple (e1,e2,e3,...,ek). For example the prime signature
> of
> > > > 340
> > > > = 2^2 * 5 * 17 is (2,1,1). A "signature leader" is the smallest n
> with
> > > > its
> > > > signature; the one with signature (2,1,1) is 60 = 2^2 * 3 * 5.
> Another
> > > > characterization of a signature leader is that its prime factors are
> > > > consecutive and the exponents never increase. The signature leaders
> are
> > > > in
> > > > OEIS at A025487 <https://oeis.org/A025487>.
> > > >
> > > > Consider a measure of by how much a number n fails to be a signature
> > > > leader: how many times would we need to replace a prime in the
> > > > factorization of n by the next smallest prime, before we reach a
> > > > signature
> > > > leader? There might be lots of chains of replacements that end in a
> > > > signature leader, but I am interested in the shortest such chain.
> > > >
> > > > For example, to "relax" 340 to a signature leader I think the minimum
> > > > number of steps is 5. One possible replacement chain is 5 -> 3, then
> 17
> > > > ->
> > > > 13 -> 11 -> 7 -> 5.
> > > >
> > > > I call this number the "signature excitation" of n. It should be 0
> for
> > > > signature leaders, and (n-1) for the nth prime. (My mental metaphor
> is
> > > > of
> > > > electrons being excited from their ground states.)
> > > >
> > > > The sequence A325799 <https://oeis.org/A325799> has the exact
> values I
> > > > expect for the signature excitation of n, but I haven't been able to
> > > > wrap
> > > > my head around the given definition. Can somebody untangle this and
> > > > assure
> > > > me that this business about the number of distinct multiset sums
> works
> > > > out
> > > > to the same thing? If so, either add a comment or tell me to do so
> and
> > > > I'll
> > > > credit you.
> > > >
> > > > --
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> > >
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> > >
> >
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> >
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