cp's OEIS Frontend

By the modularity theorem, each Q-isogeny class of elliptic curves of conductor N corresponds to a unique normalized rational weight-2 newform on Gamma_0(N). This sequence (classified by Martin & Ono) lists exactly those 12 conductors N for which that newform at level N can be written as a single eta-quotient.

a(n) is the least integer in the n-th nontrivial equivalence class under iteration of the Lagarias arithmetic derivative D. Two numbers m and n lie in the same class if repeated application of D to each, eventually produces the same value; every such class has a unique smallest member, and this sequence lists those members in ascending order.

Note that k is a fixed point of the arithmetic derivative D (i.e., D(k)=k) if and only if k=p^p for some prime p. Such one-element classes {p^p} are considered trivial and are excluded from the list of nontrivial attractors.

The values of a(n) are conjectural, contingent on the absence of further merges or the existence of nontrivial cycles beyond the computational horizon; this is analogous to Collatz dynamics.

Primes p such that Legendre(-21,p) = -1.

The volume of an n-dimensional L^p-ball of radius R is vol(R, p, n) = ((2^n)*(gamma((1/p)+1))^n /gamma((n/p)+1)*R^n for all real p > 0. For the Hilbert space L^2 we recover the familiar Euclidean volume of a unit n-ball (Pi^(n/2))/gamma((n/2)+1) which attains a maximum value at dimension n=5. The present sequence describes the same phenomenon for positive integer values of p.

From Robert L. Diersing, May 26 2020: (Start)

We can find very tight bounds for a(n):

define c := 2*gamma(1 + 1/p), then

floor(p*(c^p - 1/2) - 1/2) <= a(n) <= ceiling(p*(c^p - 1/2)).

First we treat V(n, p, 1) as a real-valued smooth function w.r.t. n, then we can maximize it. Since log(x) is an increasing function, the critical points of log(V(n, p)) will be the same as the critical points for V(n, p).

(d/dn)log(V(n, p)) = log(c) - 1/(p(digamma((n+p)/p))).

(d^2/dn^2)log(V(n, p)) = -polygamma(1, (n+p)/p))/p^2 < 0 for all n > 0, p > 0.

This is because polygamma(1, x) = Sum_{k>=1} 1/(x + k)^2 > 0 for all x > 0.

Therefore (d/dx)V(n, p) is a decreasing function w.r.t. n, which means there is only one local extremum. We can bound that local extremum similar to the way that we find bounds for the case where p=2 and the radius varies: see Wikipedia link.

Let's call this extremum n_.

dV/dn = log(c) - 1/(p*(digamma((n_+p)/p))) = 0;

p*log(c) = digamma((n_ + p) / p);

p*(inverse_digamma(plog(c)) - 1) = n_.

We can see from [2] that

log(y - 1/2) < digammma(y) < log(y + e^-m - 1),

where m is the Euler-Mascheroni constant.

Thus

log((n_+ p)/p - 1/2) < p*log(c) < log((n_ + p)/p + e^-m - 1);

c^p - e^-m + 1 < (n_ + p)/p < c^p + 1/2;

p*(c^p - e^-m) < n_ < p*(c^p - 1/2).

We can now improve on these bounds; define:

n` = p*(c^p - 1/2).

We can prove

n` - n_ < 1/2;

p*(c^p - 1/2 - inverse_digamma(p*log(c))) < p/c^p < 1/2;

c^p + 1/2 - 1/c^p < inverse_digamma(log(c^p)), say x := c^p;

x + 1/2 - 1/x < inverse_digamma(log(x));

digamma(x + 1/2 - 1/x) < log(x)

(inverse_digamma is an increasing function).

We see from [1] that digamma(1/log(1 + 1/x)) < log(x), therefore we only need to show

digamma(x + 1/2 - 1/x) < digamma(1/log(1 + 1/x)),

and since inverse_digamma is an increasing function, it suffices to show

x + 1/2 - 1/x < 1/log(1 + 1/x),

which is true for x > 0, and can be seen from an inequality solver such as Wolfram Alpha: https://www.wolframalpha.com/input/?i=x+%2B+%C2%BD+-+1%2Fx+%3C+1%2Flog%281+%2B+1%2Fx%29.

thus

n` - 1/2 < n_ < n`.

Now we have to confirm that n_ is the global maximum by checking the boundaries.

V(p, 0) = c^p < n_;

Limit_{n->oo} V(p, n) = lim_{n->oo} (2*sqrt(2*Pi/p))^n / (sqrt(2*Pi*n/p) n^n) = 0 < n_.

Therefore n_ is the global maximum, and thus we can solve for a(n).

floor(n` - 1/2) <= a(n) <= ceiling(n`).

So to find a(n) we only need to iterate over at most three values. (End)