Double and Reverse Digits

After twelve days, I encountered today the first member of the twin prime pair: 24371 and 24373. It's been a while: the last pair was 24179 and 24181 as far as I can tell. The number is a member of the interesting OEIS A036447 formed using 1 as its starting point and then doubling and reversing the digits:

1, 2, 4, 8, 61, 221, 244, 884, 8671, 24371, ...

The number is also a member of OEIS A243408: primes p such that 10p-1, 10p-3, 10p-7 and 10p-9 are all prime. This means that 243709, 243707, 243703 and 243701 are all prime.

Additionally, the number is a member of OEIS A158641: strong primes p: adding 2 to any one digit of p produces a prime number (no digits 8 & 9 in p). This means that 44371, 26371, 24571, 24391 and 24373 are prime.

There's still more. The number is a member of OEIS A104846: primes from merging of 5 successive digits in decimal expansion of e. Here is part of the sequence (up to 24371):

74713, 62497, 24977, 24709, 47093, 95957, 49669, 27427, 46639, 32003, 59921, 21817, 35729, 63073, 28627, 27943, 94349, 33829, 98807, 57383, 41879, 18793, 91499, 68477, 47741, 37423, 42437, 24371

Lastly, 24371 is also a member of OEIS A054564 as describe below:

Permutations of Digits 2 to 6

I struggled yesterday to come up with anything of significance for my 24356 tweet. I filled it out with "my prime drought is nearly over as 24359 draws nearer (previous was 24337)" but a day later I realise that I should have looked a little more closely at the number, especially in the light of my previous post. 24356 is a permutation of the digits 2, 3, 4, 5 and 6. This has occurred previously with 23456, 23465, 23546, 23564, 23645, 23654 and now with 24356, followed shortly by 24365. I need to learn to not rely solely on WolframAlpha and the OEIS for my information about my daily numbers.

Permutations of Digits 1 to 5

Even though the number 24351 doesn't seem to have much mathematical significance, it nonetheless should not pass unnoticed because the digits that comprise it are a permutation of the first five digits: 1, 2, 3, 4 and 5. The occurrences of this thus far have been:

12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 23415, 23451, 23514, 23541, 24135, 24153, 24315, 24351

Thus it can be seen that 24351 is the fortieth such occurrence and because the sum of the first five digits is 15, all of these numbers are divisible by 3 and thus none are prime. The first occurrence (12345) took place on Thursday, January 20, 1983 when I was in England with Ali and Tara. I was still 33 years of age at that time, half my current age. The continuation will be:

24513, 24531, 25134, 25143, 25314, 25341, 25413, 25431, 31245, ...

The jump from 25431 to 31245 is a big one with the latter falling on Thursday, October 19, 2034. If I make it that far, I'll be over 85 years old.

Goldbach's Conjecture and Zeckendorf's Theorem

Watching the movie A Brilliant Young Mind a couple of nights ago, I heard mention made of Goldbach's Conjecture and thought I should find out about it. This is the introduction to what Wikipedia had to say about it: 

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. It states:

Every even integer greater than 2 can be expressed as the sum of two primes.

The conjecture has been shown to hold up through 4 × 10^18, but remains unproven despite considerable effort.

The expression is not unique as the example of 100 shows:

100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53

I came across Zeckendorf's Theorem when examining the number 24331. It's similar to Goldbach's Conjecture in that it deals with the decomposition of numbers but into Fibonacci numbers, not primes. It states, to quote from Wikipedia again, that:

Every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers.

Now the first few Fibonacci numbers are:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368

The Wikipedia article goes on to state that the theorem has two parts:

Existence: every positive integer n has a Zeckendorf representation.
Uniqueness: no positive integer n has two different Zeckendorf representations.

The example is given of the number 100:

100 = 89 + 8 + 3.

There are other ways of representing 100 as the sum of Fibonacci numbers:

100 = 89 + 8 + 2 + 1

100 = 55 + 34 + 8 + 3

but these are not Zeckendorf representations because 1 and 2 are consecutive Fibonacci numbers, as are 34 and 55.

Sums of Squares

I was prompted to investigate this one when I was checking the OEIS for the prime number 24329 that arose from my daily count of the number of days that I've been breathing in Earth's air. One entry reported that 24329 is a member of sequence A100559 that lists the smallest prime equal to the sum of n distinct squares. 24329 appeared in the case of n=41 so this prime was the smallest one that was equal to the sum of 41 distinct squares. The members of the sequence, up to 24329, are:

5, 29, 71, 79, 131, 179, 269, 349, 457, 569, 719, 971, 1171, 1327, 1601, 1913, 2269, 2593, 2999, 3539, 4099, 4549, 5231, 5717, 6529, 7297, 7879, 8779, 9791, 10711, 11867, 12809, 14081, 15269, 16561, 17863, 19463, 20771, 22541, 24329

The sequence begins with 5 for the case of n=2 where \(1^2+2^2=5\). The examples shown in the OEIS entry are as follows:

\(a(3)=29 \text{ because } 29=2^2+3^2+4^2\)
\(a(4) = 71 = 1^2+3^2+5^2+6^2\)
\(a(5)=79 \text{ because } 79=1^2+2^2+3^2+4^2+7^2\)

It can be seen that the squared numbers do not necessarily need to include 1 nor all of the numbers between 1 and n. It can even include numbers greater than n. For example, 29 (when n=3) does not include 1; 71 (when n=4) begins with 1 but does not include 4 but includes 5 and 6; 79 (when n=5) includes 1, 2, 3, 4 and 7 (but not 5 or 6). 

Interestingly, if the squares of the first n numbers are added together, they do not add to a prime number for the cases n=2 to 42. It may be a mathematical fact that the sum of the squares of the integers in sequence can never add to a prime number. In the list below, I've worked out these sums in Excel and put them in the central column with the corresponding primes from OEIS A100559 in the right-hand column:

1	1
2	5	5
3	14	29
4	30	71
5	55	79
6	91	131
7	140	179
8	204	269
9	285	349
10	385	457
11	506	569
12	650	719
13	819	971
14	1015	1171
15	1240	1327
16	1496	1601
17	1785	1913
18	2109	2269
19	2470	2593
20	2870	2999
21	3311	3539
22	3795	4099
23	4324	4549
24	4900	5231
25	5525	5717
26	6201	6529
27	6930	7297
28	7714	7879
29	8555	8779
30	9455	9791
31	10416	10711
32	11440	11867
33	12529	12809
34	13685	14081
35	14910	15269
36	16206	16561
37	17575	17863
38	19019	19463
39	20540	20771
40	22140	22541
41	23821	24329
42	25585	25913

It can be seen that the primes are always larger than the consecutive sums of squares to the left but not generally larger than the next sum (except in the case n=4 where 71 is larger than 55, the sum of the first 5 squares).

Interprimes

There's always more to discover about prime numbers and numbers that are not themselves prime but are associated with them. One such set of numbers is comprised of so-called interprimes. An interprime is defined by WolframAlpha as follows:

An interprime is the average of consecutive (but not necessarily twin) odd primes. The first few terms are 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, ... (OEIS A024675). The first few even interprimes are 4, 6, 12, 18, 26, 30, 34, 42, 50, 56, 60, ... (OEIS A072568), and the first few odd ones are 9, 15, 21, 39, 45, 69, 81, 93, 99, ... (OEIS A072569).

As well as the odd and even interprimes, there are other subsets as well including the one I discovered yesterday when investigating 24323. It turns out this numbers belongs to OEIS A075288: interprimes which are of the form s*prime, s=13 e.g. 1313 is an interprime and 1313/13 = 101 is prime. The sequence runs as follows:

26, 39, 1313, 4771, 7033, 9607, 11947, 12233, 14963, 15613, 18707, 20527, 24323

The OEIS lists sequences from s = 2 up to s = 21 (A075277-A075296). So yesterday I was halfway between two prime days (24317 and 24329). So I now have a new term and a new concept in my armoury. Of interest in this regard is the following (from WolframAlpha) - doubleclick to enlarge:

Counterbalanced Numbers

 

Today's daily count turned up counterbalanced numbers that apparently have the property that:$$ \frac{\phi(n)}{\sigma(n)-n}$$is an integer. Of course, this prompted me to revisit the phi and sigma functions because I'm regularly forgetting what they signify. To recapitulate, the phi function is the Euler totient function defined by WolframAlpha as:

The totient function \(\phi(n) \), also called Euler's totient function, is defined as the number of positive integers \(\leq n\) that are relatively prime to (i.e., do not contain any factor in common with) \(n\), where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function \(\phi(n)\) can be simply defined as the number of totatives of \(n\). For example, there are eight totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so \( \phi(24)=8\).

In this case, \( \phi(24307)=23976\). The sigma function gives the number of divisors of a number and in this case \( \sigma(24307)=24640\). Thus:$$ \sigma(24307)-24307=24640-24307=333 \text{ and }\\ \frac{\phi(24307)}{333}=\frac{23976}{333}=72$$Actually the sigma function is a little more complicated because technically is comes with a subscript that can have any integer value, positive or negative. Let's say the subscript is \(k\), then the sigma function represents the sum of the \(k^{th}\) powers of the divisors. Thus when \(k=0\), it returns the number of divisors because the \(0^{th}\) powers of the divisors are 1. When \(k=1\), it returns the sum of the divisors and so it really is the \(k=1\) case that the sigma function is being used above.

on July 5th 2021