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Each octave represents a change in frequency by a factor of two. … These differ in frequency by successive factors of roughly 2 1/12 —with different temperament schemes using different rational approximations to powers of this quantity.
The total number of commutative groups with k elements is just
Apply[Times, Map[PartitionsP[Last[#]] &, FactorInteger[k]]]
(Relabelling of elements makes the number of possible operator forms up to k!
Patterns of digits in various bases generated by successive multiplication by a fixed factor.
It is known that any odd perfect number must be greater than 10 300 , must have a factor of at least 10 6 , and must be less than 4 4 s if it has only s prime factors.
The necessary transformation is the so-called Lorentz transformation
{t, x} {t - v x/c 2 , x - v t}/Sqrt[1 - v 2 /c 2 ]
And from this the time dilation factor 1/Sqrt[1 - v 2 /c 2 ] shown on page 524 follows, as well as the length contraction factor Sqrt[1 - v 2 /c 2 ] .
There are many factors which affect the details of displacements and vibrations in a solid.
With the axioms used here, the total number of strings grows by a factor of roughly 1.7 at each step; on the last steps shown there are altogether 237 and 973 strings respectively.
For prime k , each cycle (except all 0's) corresponds to a term in the product Factor[x k n - 1 - 1, Modulus k] .
But in fact only a very few other examples were found—all ultimately based on very much the same ideas as factoring. … (Factoring is not known to be NP-complete.)
And even in the case of factoring there are questions about the idealizations used.
And in fact in the case shown below there are roughly a factor 1.22 more nodes on each successive row—corresponding to overall approximate exponential growth.