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

Lucas J lucas at spicyorange.com
Mon Oct 10 15:19:32 CEST 2022

```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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