[seqfan] Re: A325799 = "Signature excitation" of n?
Allan Wechsler
acwacw at gmail.com
Tue Oct 11 05:02:20 CEST 2022
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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