The Riemann Prime Counting Function

I've been looking back over some of the entries in my Evernote notebook titled "Your Days Are Numbered" that used to be automatically posted to a blog that wanted to charge for the privilege of accessing my posts. However, the entries in the notebook remain and one that caught my attention related to the prime number 24121. OEIS A226473 states:

a(n) is the first prime index where the gap between R(n), Riemann's prime counting function, and Pi(n), the exact prime counting function, is greater than n.

The first terms in the sequence are:

109, 556, 1327, 3296, 5380, 10343, 11767, 19202, 19361, 19371, 24121, 42863 

The example given for 109 is that RiemannR(109) = 27.4664... and PrimePi(109) = 29 give the first gap greater than 1, hence a(1) = 109. 

24121 is the 11th member of the sequence. Pi(24121) = 2686 and RiemannR(24121) = 2674.9424 ... and 11 subtracted from 2686 is 2675. Thus the gap between PrimePi and RiemannR is more than 11 and so a(11) = 24121.

There is a discussion of the Riemann prime counting function on WolframAlpha.

Partitions

As I've been reading "The Man Who Knew Infinity", the topic of partitions came up and I felt impelled to delve a little further into the topic. Wikipedia defines a partition as follows:

In number theory and combinatorics, a partition of a positive integer \(n\), also called an integer partition, is a way of writing \(n\) as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways: 

4 

3 + 1 

2 + 2 

2 + 1 + 1 

1 + 1 + 1 + 1 

The order-dependent composition 1 + 3 is the same partition as 3 + 1, while the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition 2 + 1 + 1. 

A summand in a partition is also called a part. The number of partitions of \(n\) is given by the partition function \(p(n) \). So \(p(4)\) = 5. The notation \( \lambda \vdash n \) means that \( \lambda \) is a partition of \( n \). 

Partitions can be graphically visualised with Young diagrams or Ferrers diagrams.

Wolfram Alpha can be used to generate the number of partitions for a given number. An example of the number 5 is shown below, with visualisation using Ferrers diagrams:

 

The OEIS sequence for the number of partitions of the various numbers is A000041 and begins as follows:

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525

In terms of my numbered days, the last partition number was 21637 and the next will be 26015. This is only the briefest of introductions to the topic and I've not even mentioned the partition-generating function. I'll add it here just for completeness:$$\sum_{n=0}^{\infty} p(n) \, x^n=\prod_{k=1}^{\infty} \left ( \frac{1}{1-x^k} \right )$$

on Thursday April 8th 2021

Golden Semiprimes

Today I am 24534 days old and when checking this number on OEIS I came across the following:

Not having heard of the term before, I investigated what defined a golden semiprime. Below is the OEIS reference:

In today's case of 24534, the sum of the digits is 18 and the factorisation is 2×3^2×29×47, so n/(sum of digits of n) is 29 x 47. Now the absolute value of  29 x ø - 47 is approximately 0.077 and thus the criterion for a golden semiprime is satisfied. As can be seen from the list in OEIS A108542, such semiprimes are relatively sparse.

The significance of the concept is that it identifies that the fact that the two factors making up the semiprime are very nearly in the golden ratio. Of course the factors cannot be in the exact ratio because ø is not rational. Again, I've learned something quite interesting by keeping track of my daily numbers.

Diophantine Equations

Today I turned 24521 days old and it turns out that this number is a non-negative value of x in the solution (x, y) to the Diophantine equation: 

x^2+(x+16807)^2=y^2

The corresponding y value is 48055 so the solution is (24521, 48055). So I thought that this would be an appropriate time to include some information about Diophantine equations. According to Wikipedia:

The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.

 The definition given in the same article is:

In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An exponential Diophantine equation is one in which exponents on terms can be unknowns.

Today's Diophantine equation is quadratic and it's equivalent to finding integer solutions to the right-angled triangle with hypotenuse y and arms x and x + 16807. As it turns out, the requisite triangle is 48055, 24521 and 41328. In general terms, finding all right triangles with integer side-lengths is equivalent to solving the Diophantine equation a^2 + b^2 = c^2, except in this case we know that b = a + 16807.

Diophantine analysis is a big topic to which I may return at a later date but that's enough for now.

Primes Within Primes

Today is a prime day, 24517, and it turns out to be an interesting prime in that it conforms to the membership requirements of OEIS A164077 which state that it must be:

• Prime p1 of the form a^b + c^d = p1, where a, b, c, d are primes
• a + b + c + d = p2, where p2 (A164078) is prime
• conc(abcd) = p3 (concatenation of a, b, c , d) is prime (A164079)

The first such number to qualify is 5527 because:

• 5^5 + 2^7 = 3253 is prime
• 5 + 5 + 2 + 7 = 19 is prime
• conc (abcd) = 5527 is prime

The next number to qualify is 24517 because:

• 29^3 + 2^7 = 24517 is prime
• 29 + 3 + 2 + 7 = 41 is prime
• conc (abcd) = 29327 is prime

After that comes 78157 because:

• 2^5 + 5^7 = 78157 is prime
• 2 + 5 + 5 + 7 = 19 is prime
• conc (abcd) = 2557 is prime

After that comes 366103 because:

• 2^13 + 71^3 = 366103 is prime
• 2 + 13 + 71 + 3 = 89 is prime
• conc (abcd) = 213713 is prime

... and so on and so on ...

Clearly primes satisfying such stringent prime number requirements are few and far between. Here is the list as shown on the OEIS site:

3253, 24517, 78157, 366103, 548677, 705097, 1030429, 1229257, 5735467, 6438391, 12221371, 17498881, 19618243, 74084347, 118370899, 263374849, 270840151, 286199371, 410180599, 418195621, 418719781, 529483321, 565609411, 698388391

I would imagine that there are an infinity of such primes but don't ask me to prove it.

Numbers That Are Sums of Two Squares

Today I turned 24505 days old and was surprised to find the following information about this number:

In all the time that I've been analysing my daily numbers, this was the first time that I'd seen the number being representable by the sum of the squares of six different pairs of numbers. Similarly, I'd not seen a number being the hypotenuse of four different Pythagorean triples. It lead me to think whether this was a record of some sort.

Well, a little research shows that this is certainly not the case. For example, the first numbers that are representable as the sum of the squares of six different pairs of numbers are:

5525, 9425, 11050, 12025, 12325, 13325, 14365, 15725, 17225, 17425, 18785, 18850, 19825, 21125, 22100, 22525, 23725, 24050, 24505, 24650, 25925, 26650, 26825, 27625, 28730, 28925, 29725, 31025, 31265, 31450, 31525, 32045, 32825, 34450, 34645, 34850

There are numbers that are representable as the sum of the squares of seven different pairs of numbers but the first such number is 105625. Numbers that are representable as the sum of the squares of eight different pairs of numbers are interestingly far more common than seven and begin thus:

27625, 32045, 40885, 45305, 47125, 55250, 58565, 60125, 61625, 64090, 66625, 67405, 69745, 77285, 78625, 80665, 81770, 86125, 87125, 90610, 91205, 94250, 98345, 98605, 99125, 99905, 101065, 107185, 110500, 111605, 112625, 114985, 117130, 118625

71825, 93925 and 122525 are the first three numbers that are representable as the sum of the squares of nine different pairs of numbers. Of numbers that are representable as the sum of the squares of ten different pairs of numbers, the first is 138125. The OEIS doesn't give any results for 11 onwards but there's no reason to not suppose that numbers exist that are representable as the sum of the squares of eleven different pairs of numbers and beyond.

Let's summarise the results:

1. 5
2. 65
3. 325
4. 1105
5. 8125
6. 5525
7. 105625
8. 27625
9. 71825
10. 138125

I won't deal here with the numbers n for which n^2 can be represented as the sum of squares of different pairs of numbers, in other words n forms the hypotenuse of a Pythagorean triple. However, a similar analysis is possible.

Achilles Numbers, Powerful Numbers and Perfect Powers

Today I turned 24500 days old and an analysis of this number revealed that it was an Achilles number in the OEIS listings, specifically OEIS A203663: Achilles number whose double is also an Achilles number. The initial members of this sequence are listed as:

432, 972, 1944, 2000, 2700, 3456, 4500, 5292, 5400, 5488, 8748, 9000, 10584, 10800, 12348, 12500, 13068, 15552, 16000, 17496, 18000, 18252, 21168, 21296, 21600, 24300, 24500, 24696, 25000, 26136 

I'd not heard of an Achilles number before and so I investigated. Wikipedia provides this definition:

An Achilles number is a number that is powerful but not a perfect power. A positive integer \(n\) is a powerful number if, for every prime factor \(p\) of \(n\), \(p^2\) is also a divisor. In other words, every prime factor appears at least squared in the factorisation. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as \(m^k\), where \(m\) and \(k\) are positive integers greater than 1.

Now 24500 factorises to \(2^2 \times 5^3 \times 7^2\) and is thus powerful but not a perfect power. So it is an Achilles number. Multiplication by 2 does not change this because 49000 factorises to \(2^3 \times 5^3 \times 7^2\) and again powerful but not a perfect power.

Here is the Wikipedia definition of a perfect power:

A perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, \(n\) is a perfect power if there exist natural numbers \(m > 1\), and \(k > 1\) such that \(m^k = n\). In this case, \(n\) may be called a perfect \(k\)-th power. If \(k = 2\) or \(k = 3\), then \(n\) is called a perfect square or perfect cube, respectively. Sometimes 1 is also considered a perfect power (\(1^k = 1\) for any \(k\)).