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

Allan Wechsler acwacw at gmail.com
Sun Oct 9 20:40:13 CEST 2022

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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