[seqfan] Re: A325799 = "Signature excitation" of n?
Allan Wechsler
acwacw at gmail.com
Mon Oct 10 19:46:07 CEST 2022
Emmanuel Vantieghem and Lucas J, thank you for the corrections. In
particular I am embarrassed that I originally calculated the excitation of
9 to be 2 (as if it needed to be reduced to 4) rather than 1.
Perhaps I was imagining a reduction to a signature leader with the *same
signature* as the starting number. Anyway, clearly A325799 is measuring
something different, though it might be related.
I might enter a sequence or two on this theme. Any other ideas are welcome.
On Mon, Oct 10, 2022 at 9:23 AM 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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