Kempner Series

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 series \(K_d\) is a series obtained by removing all terms containing a single digit \(d\) from the harmonic series. Surprisingly, while the harmonic series diverges, all 10 Kempner series converge.

The example is given of:$$ K_1= \frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{22}+ \dots $$where it can be seen that the fractions 1/1, 1/10, 1/11 etc. have been removed from the harmonic series. In general: $$ \sum_{k=1}^{\infty} \frac{1}{k} \approx 10^n \ln{10}$$when a particular string of length \(n\) is excluded. Thus, when removing the digit 1, the result is about 23.0258509299405. It takes a lot of terms to approach this number. After one million terms, the total is only about 7.717.

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 \( |(K(m+1) - K(m)| = 1\), where \(K(m)\) = A002034(\(m\)) is the Kempner function.

This is where I first came across the word "Kempner". The initial members of the sequence are:

1, 2, 3, 4, 9, 15, 35, 63, 99, 175, 195, 483, 1443, 2057, 2115, 2299, 3363, 3843, 5082, 5475, 6723, 7865, 11235, 11913, 12005, 22747, 24963, 26978, 27555, 31683, 37635, 41514, 46255, 51075, 62464, 68643, 76704, 77283, 89375, 95874, 98595, 104975, 105412, 113398

Following the link to the Kempner function, OEIS A002034, leads to the reference to Kempner numbers:

A002034

Kempner numbers: smallest positive integer \(m\) such that \(n\) divides \(m\)!.

The initial members of this sequence are:

1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, 4, 13, 7, 5, 6, 17, 6, 19, 5, 7, 11, 23, 4, 10, 13, 9, 7, 29, 5, 31, 8, 11, 17, 7, 6, 37, 19, 13, 5, 41, 7, 43, 11, 6, 23, 47, 6, 14, 10, 17, 13, 53, 9, 11, 7, 19, 29, 59, 5, 61, 31, 7, 8, 13, 11, 67, 17, 23, 7, 71, 6, 73, 37, 10, 19, 11, 13, 79, 6, 9, 41, 83, 7

The Kempner function is also known as the Smarandache Function which Wolfram Mathworld describes as:

The Smarandache function \( \mu(n) \) is 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 \(n|\mu(n)!\) (i.e., \(n\) divides \( \mu(n)\) factorial). For example, the number 8 does not divide 1!, 2!, 3!, but does divide 4!=4·3·2·1=8·3, so \( \mu(8)=4\). 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. 

The graph of the function is interesting. See Figure 1.

Figure 1: link