Multiplicative Persistence and Multiplicative Digital Root

My diurnal age today (26748) has the property that it has a multiplicative persistence of 6. This qualifies it for membership in OEIS A199996:

A199996

Composite numbers whose multiplicative persistence is 6.

The initial members of the sequence are:

6788, 6878, 6887, 7688, 7868, 7886, 8678, 8687, 8768, 8786, 8876, 16788, 16878, 16887, 17688, 17868, 17886, 18678, 18687, 18768, 18786, 18867, 18876, 23788, 23878, 24678, 24687, 24768, 24786, 24867, 24876, 26478, 26487, 26748, 26784, 26847, 26874, 27388 ...

To quote from Wikipedia:

In number theory, the multiplicative digital root of a natural number \(n\) in a given number base \(b\) is found by multiplying the digits of \(n\) together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of \(n\). Multiplicative digital roots are the multiplicative equivalent of digital roots.

The number of iterations required to reach the multiplicative digital root is termed the multiplicative persistence. It is conjectured that there is no number with a multiplicative persistence greater than 11. The smallest numbers with multiplicative persistence of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 are:

0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899

These numbers constitute OEIS A003001 in the OEIS (see permalink for an algorithm that will generate the members of this sequence up to one million). 

Here is permalink to an algorithm that will calculate multiplicative persistence and multiplicative digital roots for a range of numbers (both composite and prime). The algorithm is easily modified to search for a specific multiplicative persistence or multiplicative digital root. Primes can be excluded by simply adding that condition to the relevant section of the code.

ADDENDUM

On October 4th 2022, my diurnal age was 26847, a permutation of the digits of 26748 and thus also having a multiplicative persistence of 6. In between these two diurnal ages, I passed 26784 days, another permutation, and 26874 is still to come. Altogether there are 120 permutations of the digits 2, 4, 6, 7 and 8. 

It's interesting to look at a breakdown of the percentages of numbers with various multiplicative persistences. Up to one million, the breakdown is:

7    0.245%

6    0.449%

5    2.47%

4    6.68%

3    12.4%

2    37.5%

1    40.3%

In this range there are no numbers with a multiplicative persistence of 8. The first such number is 2,677,889. 

Arithmetic Numbers

I was familiar with the concept of an arithmetic derivative but today I was reminded of the concept of an arithmetic number. I encounter the term almost daily when referring to the website Numbers Aplenty but hadn't paid it much attention. Here is a definition from Wikipedia:

In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is 3:$$ \frac{1+2+3+6}{4}=3$$Phrasing it more rigorously, we could say that a number \(n\) is arithmetic if the number of divisors \(d(n)\) or \( \sigma_0(n) \) divides the sum of divisors \( \sigma_1(n) \).

This ratio is not be confused with the ratio that I was investigating in my previous post on Barely Abundant Numbers. In that case, the ratio was \( \sigma(n) \) over \(n\) which determines whether a number is deficient, perfect or abundant. If the ratio is less than 2, the number is deficient. If it's greater than 2, the number is abundant. We know 6 is a perfect number because the ratio equals 2:$$ \frac{1+2+3+6}{6}=2$$The term was brought to my attention by one of the properties of the number associated with my diurnal age of 26744. The property qualities the number for membership in OEIS A107924 as the 119th member.

A107924

Even numbers \(n\) such that \(n^2\) is an arithmetic number.

296, 536, 632, 872, 1208, 1304, 1544, 2072, 2216, 2648, 2984, 3584, 3656, 3752, 3848, 3896, 3904, 3992, 4328, 4424, 4568, 4904, 5624, 5672, 5912, 6008, 6104, 6584, 6968, 7016, 7256, 7352, 7928, 8216, 8264, 8456, 8696, 8896, 8936, 9032, 9128, 9176, 9368, 9608, 9704, 10184, 10376, 10616, 10808, 11048, 11336, 11384, 11624, 12008, 12392, 12632, 12728, 12968, 13304, 13504, 13976, 14072, 14312, 14408, 14472, 14648, 14984, 15512, 15704, 15992, 16088, 16424, 16568, 16616, 16664, 16952, 17064, 17096, 17432, 17768, 18008, 18056, 18344, 18536, 18776, 19016, 19112, 19592, 19784, 19832, 20024, 20072, 20456, 20888, 21368, 21464, 21608, 21704, 22136, 22376, 22808, 22952, 23048, 23384, 23488, 23872, 24152, 24392, 24488, 24776, 25496, 25592, 25736, 25832, 26072, 26168, 26408, 26504, 26744

The OEIS comments state that "odd numbers with this property are much more numerous" and this is indeed true. In the same range (up to 26744) there are 2462 such odd numbers. Checking for \(n=26744\), we find that \( n^2=715241536 \) and we have: $$ \frac{\sigma(715241536,1)}{\sigma(715241536, 0)}=\frac{1419732111}{21}=67606291$$It can be noted that 26744 is itself an arithmetic number but this is hardly surprising as most numbers are and this includes all odd primes. To see this, take any odd prime \(p\) with the sum of its two divisors being \(1+p\), an even number and thus divisible by 2. According to Numbers Aplenty, \(p^p\) is also an arithmetic number.

It is possible to find quite long runs of consecutive arithmetic numbers. For example, one of length 105 starts at 3033935561. The smallest 3 × 3 magic square made of consecutive arithmetic numbers is shown below:

The 10000th arithmetic number is 12953. This gives a good idea of their frequency and why numbers that aren't arithmetic numbers are more interesting because they are less frequent. For example, \(84=2^2 \times 3 \times 7\) has twelve divisors 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 that sum to 224. It is not an arithmetic number because 12 does not divide 224 without remainder.

Just to confuse matters, \(26744 = 2^3 \times 3343\) is a tau number, defined as a number that is divisible by the number of its divisors (in this case, eight). If we put all this together and ask what even numbers \(n\) have the properties that:$$ \begin{align} \frac{ \sigma(n^2,1)}{\sigma(n^2,0)}&=k_1\\ \frac{ \sigma(n,1)}{\sigma(n,0)}&=k_2\\ \frac{ n}{\sigma(n,0)}&=k_3 \end{align}$$where \(k_1, k_2, k_3\) are integers. It turns that the list is fairly short in the range up to 26744: 

632, 1208, 3896, 6008, 6584, 7352, 7928, 8696, 10616, 11384, 13304, 14072, 14648, 15992, 20024, 21368, 22136, 26168, 26744

Barely Abundant Numbers

Today's diurnal age of 26742 days threw up an interesting number property that I can't recall coming across before. This is not surprising because the last occurrence occurred when I was 17816 days old, long before I started keeping track of the numbers associated with my diurnal age. The property in question qualifies both numbers for inclusion in OEIS A071927:

A071927

Barely abundant numbers: abundant \(n\) such that \( \dfrac{\sigma(n)}{n }< \dfrac{\sigma(m)}{m} \)for all abundant numbers \(m<n,\) \( \sigma(n) \) being the sum of the divisors of \(n\).

The terms in the sequence up to one million are:

12, 18, 20, 70, 88, 104, 464, 650, 1888, 1952, 4030, 5830, 8925, 17816, 26742, 26778, 26886, 26898, 26958, 27042, 27078, 27102, 27114, 27138, 27282, 27294, 27366, 27402, 27498, 27546, 27582, 27618, 27726, 27822, 27834, 27858, 27894, 27906, 27942, 27978, 28038, 28074, 28146, 28218, 28326, 28338, 28374, 28398, 28506, 28554, 28698, 28722, 28734, 28758, 28794, 28806, 28878, 28902, 28986, 29166, 29226, 29262, 29334, 29418, 29454, 29514, 29586, 29598, 29622, 29658, 29706, 29742, 29802, 29814, 29838, 29922, 29958, 29994, 30018, 30054, 30066, 30126, 30138, 30234, 30306, 30354, 30462, 30486, 30522, 30594, 30606, 30642, 30678, 30714, 30882, 30918, 31002, 31026, 31074, 31134, 31182, 31254, 31362, 31386, 31398, 31422, 31566, 31638, 31674, 31686, 31782, 31818, 31854, 31938, 31998, 32082, 32106, 32128, 77744, 91388, 128768, 130304, 442365, 521728, 522752

Figure 1 shows a plot of these values using a vertical log axis and the long evenly spaced stretch from 26742 to 32128 stands out clearly.

Figure 1: permalink

Figure 2 shows the same numbers showing the sigma(n)/n ratios and the factorisation (click on the image to enlarge):

Figure 2: permalink

It can be seen that all the barely abundant numbers from 26742 to 32128 are sphenic numbers of the form 2 x 3 x prime. Interesting all these numbers are admirable numbers, that is numbers whose proper factors add to the number when one of the factors is made negative. With these numbers the factor to be made negative is always 6. For example, the factors 1, 2, 3, -6, 4457, 8914, 13371  add to the admirable number 26742 and the factors 1, 2, 3, -6, 5039, 10078, 15117 add to the admirable number 30234.

It's easy to see why the 2 x 3 x prime are so successful in forming barely abundant numbers. The divisors of a number \(n\) in that case are \(2, 3, n/6, n/3, n/2, n\) and the sum of these divisors is \(2n+5\), giving a \( \sigma(n)/n \) ratio of \(2+5/n \). As \(n\) gets larger, the ratio gets smaller as can be seen in the progressive decrease in the size of the ratio from 26742 to 32128. Why the abrupt gap occurs after 32128 I'm not sure.

So barely abundant numbers will be cropping up quite regularly for the next six to seven years until another "drought" of such numbers occurs between 32129 and 77743.

Squarefree Semiprime Chains

The number associated with my diurnal age today, 26734, has an interesting property that qualifies it for membership in OEIS A177215:

A177215 Numbers \(k\) that are the products of two distinct primes such that \(2k-1, 4k-3, 8k-7, 16k-15, 32k-31 \text{ and } 64k-63\) are also products of two distinct primes.

Let's confirm that:

There are 281 members of OEIS  A177215 up to one million. I'll use the highest of them, 998185, as an example. It has the properties shown below:

What if we raise the stakes so to speak and apply the additional criterion that \(128k-123\) is also a semiprime with two distinct prime factors (in other words it's squarefree)? Well, the number shrinks to 81 in the range up to one million. The highest number that qualifies in that range is 991237 with the following properties:

Let's push thing further. How many numbers qualify once we add the additional criterion of \(256k-255\) being a squarefree semiprime? The number now reduces to 29, the highest of which remains our 991237. The revised table is shown below:

Let's keep pushing. How many numbers quality if we add the additional criteria that \(512k-511\) must be a squarefree semiprime? The number is now down to 14 and 991237 is still at the top. Here's the revised table.

Pushing further to include \(1024k-1023\), we find that only four numbers qualify in the range up to one million. These are 173311, 346621, 464245 and 563326. The table below shows the properties for 563326:

Proceeding logically, we now look at the criterion \(2056k-2055\) and we find that there is only one man left standing and that is 173311. Its table properties is shown below:

So overall, an interesting journey that ends with 173311 because this number doesn't pass the \(4096k-4095\) test. Of course, the journey never ends. We must ask what is the smallest number greater than one million that satisfies the criterion that \(4096k-4095\) is a squarefree semiprime? Well that number is 2212801. It's properties are shown below:

I'll stop there but here is a SageMathCell permalink to the program that I was using to discover these numbers. Feel free to experiment further.

ADDENDUM:

After making this post, I discovered that I'd covered much of the same territory in a post from October 17th 2018 titled Semiprime Chains. It will teach me to look back over my previous posts before posting. I've made so many posts of the years that it's easy to forget my earlier posts. One thing that is different between the two posts is the code that I used. The latter code is far more succinct. It's just as well that I looked back because the embedded code that I'd used was outdated. It contained the superseded print \(n\) command instead of the Python 3 print(\(n\)) command. I've corrected that now.

Here are the links to my previous posts on semiprimes:

• Brilliant Numbers
• Pandigital Numbers Formed From the Product of a Number and its Reversal
• Golden Semiprimes
• An Unhappy Family
• A Prime to Remember
• Semiprime Factor Ratios
• More about Golden Semiprimes
• 2019: A Numerical Profile
• Semiprime Chains
• Another Look at Semiprimes
• Visualisation of Semiprimes
• Semiprimes that Approximate Whole Numbers

I'm surprised at how many posts I've made over the years about semiprimes. It would be interesting to make a small booklet on the topic given the amount of material that I've accumulated.

My Yearly Pronic Number

Pronic numbers are numbers of the form \(n \times (n+1) \) where \(n\) is an integer \( \geq 1\). Thus the first such number is 2. Here are the pronic numbers up to 40,000:

2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660, 3782, 3906, 4032, 4160, 4290, 4422, 4556, 4692, 4830, 4970, 5112, 5256, 5402, 5550, 5700, 5852, 6006, 6162, 6320, 6480, 6642, 6806, 6972, 7140, 7310, 7482, 7656, 7832, 8010, 8190, 8372, 8556, 8742, 8930, 9120, 9312, 9506, 9702, 9900, 10100, 10302, 10506, 10712, 10920, 11130, 11342, 11556, 11772, 11990, 12210, 12432, 12656, 12882, 13110, 13340, 13572, 13806, 14042, 14280, 14520, 14762, 15006, 15252, 15500, 15750, 16002, 16256, 16512, 16770, 17030, 17292, 17556, 17822, 18090, 18360, 18632, 18906, 19182, 19460, 19740, 20022, 20306, 20592, 20880, 21170, 21462, 21756, 22052, 22350, 22650, 22952, 23256, 23562, 23870, 24180, 24492, 24806, 25122, 25440, 25760, 26082, 26406, 26732, 27060, 27390, 27722, 28056, 28392, 28730, 29070, 29412, 29756, 30102, 30450, 30800, 31152, 31506, 31862, 32220, 32580, 32942, 33306, 33672, 34040, 34410, 34782, 35156, 35532, 35910, 36290, 36672, 37056, 37442, 37830, 38220, 38612, 39006, 39402, 39800

I've marked the pronic number 26732 = 163 x 164 in bold because that is my diurnal age today (June 11th 2022) and this fact is what prompted me to make this post. The previous such number (26406 = 162 x 163) occurred on Tuesday, July 20th 2021 and the next (27060 = 164 x 165) will occur on Friday, May 5th 2023. So at the moment, a pronic number appearing as my diurnal age is pretty much a yearly thing and as such should be celebrated.

Pronic numbers are also called oblong numbers, rectangular numbers or heteromecic numbers. Interestingly, the sum of the reciprocals of the pronic numbers is 1. Thus:$$\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=1$$I've written about numbers of this sort before in a post titled Pronic Pandigital Numbers and Beyond on July 23rd 2021. Over 80% of pronic numbers are abundant but 26732 is deficient. In fact, of the 199 numbers in the list above, only 35 are deficient. These are:

2, 110, 182, 506, 1406, 1892, 2162, 2756, 3422, 3782, 4556, 5402, 6806, 7310, 8930, 9506, 11342, 11990, 14042, 14762, 17030, 17822, 18632, 20306, 21170, 22052, 22952, 24806, 26732, 27722, 29756, 31862, 32942, 36290, 37442

This sequence of numbers forms part of OEIS A077804:

A077804

Deficient oblong numbers.

The generating function for the pronic numbers is:$$\frac{2x}{(1-x)^3}=2x+6x^2+12x^3+20x^4+ \dots$$Pronic numbers are also figurate numbers of the form:$$P_n=2T_n=n(n+1)$$where \(T_n\) is the \(n^{th}\) triangular number. A very few pronic numbers are palindromic. The first few are listed below:

2, 6, 272, 6006, 289982, 2629262, 6039306, 27999972, 28233282, 2704884072, 20278187202, 20591819502, 2592587852952, 2936231326392, 21809166190812, 27237788773272, 229145919541922, 233552101255332, 250087292780052, 2243922442293422, 2570769009670752, 20333113431133302, 27785925652958772

These numbers form OEIS A028337:

A028337

Palindromes of the form n(n+1).