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

David Corneth davidacorneth at gmail.com
Mon Oct 10 16:55:18 CEST 2022

```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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>

```