I'm trying to tabulate the first 8
colossally abundant numbers
defined by critical epsilon values, with epsilon listed in descending order (Broughan lists the first few here, although note that the errata for his book here add one missing critical epsilon value, which I have included in the list below). I'm listing them in TeXForm
form simply because it's easier to decipher visually. However, the same values are already entered in the Mathematica code that follows. Those epsilon values are
$\epsilon_1=\frac{\log \left(\frac{3}{2}\right)}{\log (2)}$, $\epsilon_2=\frac{\log \left(\frac{4}{3}\right)}{\log (3)}$, $\epsilon_3=\frac{\log \left(\frac{7}{6}\right)}{\log (2)}$, $\epsilon_4=\frac{\log \left(\frac{6}{5}\right)}{\log (5)}$, $\epsilon_5=\frac{\log \left(\frac{15}{14}\right)}{\log (2)}$, $\epsilon_6=\frac{\log \left(\frac{13}{12}\right)}{\log (3)}$, $\epsilon_7=\frac{\log \left(\frac{8}{7}\right)}{\log (7)}$, $\epsilon_8=\frac{\log \left(\frac{31}{302}\right)}{\log (2)}$
I compose these into a table...
Quiet[
xList = Range[8];
caNumbers = {2, 6, 12, 60, 120, 360, 2520, 5040};
epsilon =
Table[ϵ, {ϵ, {Log[3/2]/Log[2], Log[4/3]/Log[3],
Log[7/6]/Log[2], Log[6/5]/Log[5], Log[15/14]/Log[2], Log[13/12]/Log[3],
Log[8/7]/Log[7], Log[31/30]/Log[2]}}];
epsilonN =
Table[N[ϵ], {ϵ, {Log[3/2]/Log[2], Log[4/3]/Log[3],
Log[7/6]/Log[2], Log[6/5]/Log[5], Log[15/14]/Log[2], Log[13/12]/Log[3],
Log[8/7]/Log[7], Log[31/30]/Log[2]}}];
colossalNumber =
Table[N[Product[p^(Floor[ Log[(p^(1 + ϵ) - 1)/
(p^ϵ - 1)]/Log[p]] - 1),
{p, Prime[Range[25]]}]], {ϵ, {Log[3/2]/Log[2], Log[4/3]/Log[3],
Log[7/6]/Log[2], Log[6/5]/Log[5], Log[15/14]/Log[2], Log[13/12]/Log[3],
Log[8/7]/Log[7], Log[31/30]/Log[2]}}];
criticalEpsilonValueTable =
Transpose[
Join[{xList, epsilon, epsilonN, N[colossalNumber], caNumbers}, 2]];
TableForm[criticalEpsilonValueTable,
TableHeadings -> {None, {"x", "ϵ", "ϵ (numeric)",
"\!\(\*SubscriptBox[\(n\), \(ϵ\)]\) ∈ CA",
"xth CA number"}}]
]
...and the value of n
for x = 3
is wrong. In fact, the third CA number is missing entirely:
Larger tables produce similar results - missing values of n
.
It's possible I've made some mathematical error; I am aware that there is more than one CA number associated with each critical value of epsilon. But the numbers listed under xth CA number
are the lowest of each set... So, I wonder if it's to do with machine precision?
Does anyone have any suggestions?