Sums of Digits

The number associated with my diurnal age today, 28248, has the property that its digits raised to the fourth power are equal to its totient.$$ \begin{align} 2^4 + 8^4 + 2^4 + 4^4 + 8^4 &= 8480 \\ \phi(28248) &= 8480 \end{align}$$This qualifies it for membership in OEIS A269669:

A269669 numbers whose Euler totient function is equal to the sum of some fixed power of their digits.

The numbers in this sequence are few and far between, with 28248 being the last in the range up to 40000. The initial members are 1, 2, 20, 40, 228, 352, 712, 813, 835, 2079, 4020, 28248. Take 20 as another example:$$ \begin{align} 2^3 + 0^3 &= 8 \\ \phi(20) &= 8 \end{align}$$As part of my daily number analysis, I consider:

• the sum of the digits of a number
• the sum of the digits squared of a number
• the sum of the digits cubed of a number

If all three sums are prime, then I make a note of it. I consider higher powers of the digits in the context of narcissistic numbers where a narcissistic number is defined as a \(k\)-digit nonnegative number equal to the sum of the \(k\)-th powers of its digits. An example is of such a number is 9474 where:$$9474=9^4+4^4+7^4+4^4$$D-powerful numbers are similar but are defined as integers that can be expressed as a sum of positive powers of their digits. An example is 994 where:$$994=9^3+9^1+4^4$$OEIS A269669 relates such sums to a number's totient which I hadn't thought of doing before. This idea can be extended to other number properties of a number and I've included a list of such properties as output from my daily number analysis. The example below is for 28248 where the equality between its totient and the sum of its digits raised to the fourth power can be clearly seen.

  Daily Number                      28248
  Sum of Divisors                   77760
  Sum of Proper Divisors            49512
  Totient                           8480
  Sum of DISTINCT prime factors     123
  Sum of Unitary Divisors           46656
  Sum of Digits                     24
  Sum of Digits Squared             152
  Sum of Digits Cubed               1104
  Sum of Digits to Fourth Power     8480
  Sum of Digits to Fifth Power      66624
  Product of Digits                 1024
  Gray Code                         22900
  Binary Complement                 4519
  Arithmetic Derivative             54620
  Determinant of Circulant Matrix   11904

If we search for numbers whose arithmetic derivative is equal to the sum of some fixed power of their digits, we don't find many. In the case of \(n=2\), we only have 581, 8549 and 16999 with sums of digits squared and arithmetic derivatives of 90,186 and 280 respectively in the range up to 40000. For \(n=3\) we only have 142, 6127 and 12643 with sums of digits cubed and arithmetic derivatives of 73, 568 and 316 respectively. For higher powers, nothing in the range up to 40000. Here is the permalink.$$ \begin{align} 581 &\rightarrow 5^2+8^2+1^2 = 90 \text{ = arithmetic derivative of 581} \\ 8549 &\rightarrow 8^2+5^2+4^2+9^2 = 186 \text{ = arithmetic derivative of 8549} \\ 16999 &\rightarrow 1^2+6^2+9^2+9^2+9^2 = 280 \text{ = arithmetic derivative of 16999} \\ 142 &\rightarrow 1^2+4^3+2^3 = 73 \text{ = arithmetic derivative of 142} \\ 6127 &\rightarrow 6^3+1^3+2^3+7^3 = 568 \text{ = arithmetic derivative of 6127} \\ 12643 &\rightarrow 1^3+2^3+6^3+4^3+3^3 =316 \text{ = arithmetic derivative of 12643} \end{align}$$

An Interesting Triple 7 Number

Today I turned \( \textbf{27717} \) days old and this number has a plethora of interesting properties that deserve a special mention and thus a dedicated post. Here are some of those properties.

• \( \textbf{27717} \) is a so-called Lucky Cube, meaning it is a number whose cubes contain the digit sequence “888”, here:$$27717^3 = 21293088810813$$The numbers that satisfy from 27717 to 40000 are: 27717, 27942, 27973, 28192, 28442, 28484, 28692, 28740, 28942, 29079, 29192, 29354, 29387, 29391, 29418, 29420, 29442, 29491, 29642, 29692, 29942, 29989.
  
  
• \( \textbf{27717} \) is the lesser of a pair of adjacent composite numbers such that both are only one step away from their home primes. Here: 
• \(27717 = 3 \times 9239 \rightarrow 39239\)
• \(27718 = 2 \times 13859 \rightarrow 213859\)
  
  
• \( \textbf{27717} \) is a number such that n + POD(n) and n - POD(n) are both prime (where POD stands for Product Of Digits). Here we have POD = 686:
• \(27717 + 686 = 28403\) which is a prime number
• \(27717 - 686 = 27031\) which is a prime number
  
  
• \( \textbf{27717} \) is an interprime number because it is at equal distance from the previous prime (27701) and the next prime (27733).
  
  
• \( \textbf{27717} \) is a number whose sum of divisors has prime factors (ignoring multiplicity) that multiply to the factorial 2310 where
  
  \(2310= 2 \times 3 \times 5 \times 7 \times 11\)
  
  Here 27717 has a sum of divisors 36960 and
  
  \(36960= 2^5 \times 3 \times 5 \times 7 \times 11\)
  
  but also forms a consecutive pair with 27718 because its sum of the divisors is 41580 and
  
  \(41580= 2^2 \times 3^3 \times 5  \times 7 \times11\)
  
  See blog post Primorials and the Sigma Function.
  
  
• \( \textbf{27717} \) is the TENTH member of an interesting number chain (which is base independent):
  
• \(27708 = 12 \times 2309\)
• \(27709 = 11 \times 2519\)
• \(27710 = 10 \times 2771\)
• \(27711 = 9 \times 3079\)
• \(27712 = 8 \times 3464\)
• \(27713 = 7 \times 3959\)
• \(27714 = 6 \times 4619\)
• \(27715 = 5 \times 5543\)
• \(27716 = 4 \times 6929\)
• \(27717 = 3 \times 9239\)
• \(27718 = 2 \times 13859\)
  

See blog post Count Down Number Chains  

 

• \( \textbf{27717} \) is a cyclic number.
  
  
• \( \textbf{27717} \) is a xenodrome in base 9 : 42016. See blog post Xenodromes.
  
  
• \( \textbf{27717} \) is a number that does not reach a palindrome after 2001 cycles of the reverse and add algorithm.
  
  
• \( \textbf{27717} \) is a D-number meaning it is a number \(n > 3\) such that n divides \( k^{n-2}- k\) for all \(1 < k < n\) relatively prime to \(n\).
  
  
• \( \textbf{27717} \) can be rendered as a digit equation as follows: \(2 - \dfrac{7}{7} = 1 ^ 7\)

Carmichael Numbers

Before discussing Carmichael numbers, the prelude to my interest in these numbers must be described. On the 29th January 2019, I turned 25503 days old. As always, I started my morning by investigating the mathematical properties of this number, looking firstly for relevant entries in the Online Encyclopaedia of Integer Sequences or OEIS. Nothing much of interest showed up so I moved on to Numbers Aplenty where mention was made that it's a D-number.

The description for a \(D\)-number was:

Also known as \(3\)-Knödel numbers, they are numbers \(n>3\) such that \(n\) divides \(k^{n-2}-k\) for all \(1<k<n\) relatively prime to \(n\). For example, \(9\) is a D-number since it divides all the numbers  \(2^7-2\),  \(4^7-4\), \(5^7-5\), \(7^7-7\) and \(8^7-8\).

D-numbers are listed in the OEIS (A033553) but 25503 does not show up in a search because it's too far down the list of numbers. Only the following numbers are displayed:

9, 15, 21, 33, 39, 51, 57, 63, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 195, 201, 213, 219, 237, 249, 267, 291, 303, 309, 315, 321, 327, 339, 381, 393, 399, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 693, 699, 717, 723, 753, 771, 789, 807, 813, 819

All of these numbers are odd and composite, with most being divisible by 3. The first term that isn't divisible by 3 is 50963, the 2000\(^{th}\) term. 25503 is the 1092\(^{nd}\) term with neighbours 25401 and 25539.

Just to confuse matters, there are two ways of expressing the condition for a number to be a \(D\)-number or \(3\)-Knödel number. This is because:$$ \frac{k^{n-2}-k}{n}=k \times \frac{k^{n-3}-1}{n} $$We know that \(k\) and \(n\) are coprime, so \(n\) must divide \( k^{n-3}-1 \). This means that:$$\begin{align}
k^{n-3}-1&\equiv 0 \pmod{n} \\
k^{n-3} &\equiv 1 \pmod{n}

\end{align} $$Thus the 3 in the \(3\)-Knödel designation comes from the \(n-3\) index to the \(k\) base and the generalisation follows that a \(n\)-Knödel number for a given positive integer \(n\) is a composite number \(m\) with the property that each \(k < m\) coprime to \(m\) satisfies \( k^{m-n} \equiv 1 \pmod {m}\). The concept is named after Walter Knödel. The set of all \(n\)-Knödel numbers is denoted \(K_n\). The special case \(K_1\) represents the Carmichael numbers.

As can be seen by the initial values in Figure 1, the Carmichael numbers are few and far between:

Figure 1: source 

The Carmichael numbers lead us on to Fermat's Little Theorem which states that if \(p\) is a prime number and \(a\) is a natural number then:$$a^{\,p-1}-1 \equiv 0 \pmod{p} $$There is a proof of this theorem by mathematical induction on WolframMathWorld and the theorem shows that:

if p is prime, there does not exist a base \(a<p\) with \(a\) and \(p\) coprime such that \(a^{\,p-1}-1\) possesses a nonzero residue modulo \(p\). If such base \(a\) exists, \(p\) is therefore guaranteed to be composite. However, the lack of a nonzero residue in Fermat's little theorem does not guarantee that \(p\) is prime. The property of unambiguously certifying composite numbers while passing some primes make Fermat's little theorem a compositeness test which is sometimes called the Fermat compositeness test. A number satisfying Fermat's little theorem for some nontrivial base and which is not known to be composite is called a probable prime. Composite numbers known as Fermat pseudoprimes (or sometimes simply "pseudoprimes") have zero residue for some values of \(a\) and so are not identified as composite. Worse still, there exist numbers known as Carmichael numbers (the smallest of which is 561) which give zero residue for any choice of the base \(a\) relatively prime to \(p\). 

This brings us full circle and though much more could be said, at least I have a firmer grasp of what Carmichael numbers are all about. I have written about Fermat pseudoprimes in an earlier post on Thursday, 30 August 2018. I also make reference to the Carmichael numbers in that post, noting that these numbers have at least three prime factors e.g. 561 = 3 x 11 x 17.

Before finishing up, I should refer to the OEIS A002997 entry for the Carmichael numbers. These are the numbers listed there (note that the majority end in the digit 1):

561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461  

ADDENDUM (added September 7th 2021):

A225509

-5-Knödel numbers.

15, 55, 75, 91, 175, 247, 275, 715, 775, 1275, 1435, 2275, 2635, 3075, 3355, 4615, 6355, 6475, 7975, 8827, 9139, 10075, 10675, 11275, 11935, 13515, 14555, 21775, 26455, 28975, 30415, 31675, 32395, 43615, 46075, 47275, 52195, 59755, 64255, 77275, 78403, 81055

An interesting extension of \(n\)-Knödel numbers to \(n\) negative, in this case \(n= -5\). Composite numbers \(m > 0\) such that if \(1 < a < m\) and gcd(\(m,a\)) = 1 then \(a^{\,m+5} \equiv 1 \pmod {m}\). Permalink.