I came upon this series quite by accident when investigating what are called Kempner numbers and the Kempner function. Both of these are quite different to the Kempner series defined by Wolfram Mathworld as follows:
A Kempner seriesis a series obtained by removing all terms containing a single digit from the harmonic series. Surprisingly, while the harmonic series diverges, all 10 Kempner series converge.
The example is given of:
Another example is given of removing the string "314" which gives an approximation of 2302.58509299405 (just move the decimal place two to the right) whereas the actual result is closer to 2299.829782. Quite a good approximation. This convergence is surprising given that removing all the composite numbers in the harmonic series still results in its divergence.
So what about Kempner numbers and the Kempner function? The number associated with my diurnal age today, 26978, has a property that qualifies it for inclusion in OEIS A346211:
A346211 | Numbers m such that |
A002034 | Kempner numbers: smallest positive integer |
The Smarandache functionis the function first considered by Lucas (1883), Neuberg (1887), and Kempner (1918) and subsequently rediscovered by Smarandache (1980) that gives the smallest value for a given n at which (i.e., divides factorial). For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!=4·3·2·1=8·3, so . This function has the property that it has a highly inconsistent growth rate: it grows linearly on the prime numbers but only grows sublogarithmically at the factorial numbers.
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Figure 1: link |
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