Rhonda Numbers Revisited

I first reviewed Rhonda Numbers in an eponymous post on the 16th October 2018. Read that post to find out what defines such numbers. Today, upon turning 28280 days old, I was reminded of them again because this number is a member of OEIS A255731:

A255731: Rhonda numbers in sexagesimal number system.

The program that I wrote back then only covered bases from 2 to 36 so I got Gemini to write a new SageMath program that would accommodate every base. Here is what is generated as output when asked to find all Rhonda numbers in base 60 in the range up to one million (permalink):

3348, 3510, 6750, 17430, 18750, 18876, 18944, 19475, 20564, 21312, 26550, 28280, 37230, 38396, 43940, 48042, 77770, 88270, 91224, 97470, 108882, 111403, 120046, 123630, 181996, 182646, 235467, 253460, 260429, 264735, 278675, 289161, 295960, 296055, 306642, 324394, 325593, 337040, 348641, 361221, 377130, 378444, 398274, 411342, 412930, 441048, 444405, 450528, 453470, 458136, 469098, 499533, 503310, 517803, 533731, 534795, 541807, 547515, 566754, 598695, 612374, 612870, 626535, 630410, 656370, 656750, 667491, 670548, 684456, 701765, 703304, 705256, 706275, 709475, 720279, 750225, 757576, 762745, 765245, 809107, 812658, 821106, 877300, 880045, 881454, 915348, 927303, 929830, 930304, 936573, 936675, 967509, 972196, 973549, 984485, 986895, 998430

I also realised that I hadn't included the identification of Rhonda numbers in my daily number analysis and that deficiency has now been remedied. Here the output of the program that I got Gemini to write for the input 29280 (permalink):

28280 : determination of whether it is a Rhonda number in a given base

--- Evaluating 28280 in Base 60 ---

1. Base-60 Digits: [7, 51, 20]

2. Digit Product: 7 * 51 * 20 = 7140

3. Prime Factorization: 2^3 * 5 * 7 * 101

4. Sum of Factors: 2 + 2 + 2 + 5 + 7 + 101 = 119

5. Target Equation (Base * Sum): 60 * 119 = 7140

RESULT: True. 28280 IS a Rhonda number in base 60.

I've set the range of bases to be investigate to 100,000 so that even a number like 889200 with is a Rhonda number in TEN different bases has an accurate output:

--- Evaluating 889200 in Base 1512 ---

1. Base-1512 Digits: [588, 144]

2. Digit Product: 588 * 144 = 84672

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 1512 * 56 = 84672

RESULT: True. 889200 IS a Rhonda number in base 1512.

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--- Evaluating 889200 in Base 2760 ---

1. Base-2760 Digits: [322, 480]

2. Digit Product: 322 * 480 = 154560

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 2760 * 56 = 154560

RESULT: True. 889200 IS a Rhonda number in base 2760.

-----------------------------------

--- Evaluating 889200 in Base 5160 ---

1. Base-5160 Digits: [172, 1680]

2. Digit Product: 172 * 1680 = 288960

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 5160 * 56 = 288960

RESULT: True. 889200 IS a Rhonda number in base 5160.

-----------------------------------

--- Evaluating 889200 in Base 7904 ---

1. Base-7904 Digits: [112, 3952]

2. Digit Product: 112 * 3952 = 442624

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 7904 * 56 = 442624

RESULT: True. 889200 IS a Rhonda number in base 7904.

-----------------------------------

--- Evaluating 889200 in Base 9400 ---

1. Base-9400 Digits: [94, 5600]

2. Digit Product: 94 * 5600 = 526400

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 9400 * 56 = 526400

RESULT: True. 889200 IS a Rhonda number in base 9400.

-----------------------------------

--- Evaluating 889200 in Base 10032 ---

1. Base-10032 Digits: [88, 6384]

2. Digit Product: 88 * 6384 = 561792

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 10032 * 56 = 561792

RESULT: True. 889200 IS a Rhonda number in base 10032.

-----------------------------------

--- Evaluating 889200 in Base 11440 ---

1. Base-11440 Digits: [77, 8320]

2. Digit Product: 77 * 8320 = 640640

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 11440 * 56 = 640640

RESULT: True. 889200 IS a Rhonda number in base 11440.

-----------------------------------

--- Evaluating 889200 in Base 12920 ---

1. Base-12920 Digits: [68, 10640]

2. Digit Product: 68 * 10640 = 723520

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 12920 * 56 = 723520

RESULT: True. 889200 IS a Rhonda number in base 12920.

-----------------------------------

--- Evaluating 889200 in Base 14136 ---

1. Base-14136 Digits: [62, 12768]

2. Digit Product: 62 * 12768 = 791616

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 14136 * 56 = 791616

RESULT: True. 889200 IS a Rhonda number in base 14136.

-----------------------------------

--- Evaluating 889200 in Base 15080 ---

1. Base-15080 Digits: [58, 14560]

2. Digit Product: 58 * 14560 = 844480

3. Prime Factorization: 2^4 * 3^2 * 5^2 * 13 * 19

4. Sum of Factors: 2 + 2 + 2 + 2 + 3 + 3 + 5 + 5 + 13 + 19 = 56

5. Target Equation (Base * Sum): 15080 * 56 = 844480

RESULT: True. 889200 IS a Rhonda number in base 15080.

-----------------------------------

A Timely Surprise

When searching for interesting properties associated with the number (27491) representing my diurnal age, I encountered OEIS A335789:

A335789

a(\(n\)) = time to the nearest second at the \(n\)-th instant (n>=0) when the hour and minute hands on a clock face coincide, starting at time 0:00.

The initial members of the sequence are: 0, 3927, 7855, 11782, 15709, 19636, 23564, 27491, 31418, 35345, 39273, 43200, 47127, 51055, 54982, 58909, 62836, 66764, 70691, 74618, 78545, 82473, 86400, 90327, 94255, 98182, 102109, 106036, 109964, 113891, 117818, 121745, 125673, 129600, 133527, 137455, 141382

Clock face after 27491 seconds from midnight

The comment is made that "after 12 hours or 43200 seconds, the hands overlap at 12:00 and the cycle repeats". What surprised me was that I was the author of this sequence, approved on August 14th of 2020, under the pseudonym of Sean Lestrange. I wrote about this sequence and related matters in a blog post titled Sexagesimal Number System on the same date. I'd quite forgotten about until I received this pleasant reminder.

Of course I no longer propose new sequences to the OEIS for reasons I've explained but I continue to make daily use of it. It's not the enormous body of sequences contained in the OEIS that I have a problem with. Instead it's the people who control the approval process.

This number of seconds corresponds to 7 hours 38 minutes 11 seconds. Figure 1 shows the clock's appearance for all those times when the hour and minute hands of the clock coincide (source).

It will be quite a while before the hour and minute hands of the clock overlap again when reckoning one second of clock time equal to one day of my lifetime. Tuesday, April 10th 2035 to be precise.

Sexagesimal Number System

Today I turned 26066 days old and the OEIS had little of interest to say about the properties of this number. It was the same with Numbers Aplenty, my other source of information about number properties. The latter site did remind me that the number could be expressed as a sum of two squares, \(71^2+145^2 \), with my age in years (71) appearing coincidentally. I searched for some time on other sites to find something else of interest but could find nothing.

Finally, I thought about the sexagesimal or base 60 number system. What if I interpreted 26066 as a time in seconds and converted it to its equivalent in hours, minutes and seconds? Doing this, the result is 7:14:26. As I wrote in the graphic (Figure 1) and which I'll reproduce here in case of difficulty in reading it:

Using 1 second equals 1 day, it takes about two months for the minute hand to progress one tick and 120 months or about 10 years for the hour hand to progress five ticks and mark out the passage of an hour. The hours thus mark out the decades and the minutes mark the passage through the months and years of that decade. It’s an interesting way of viewing the passing of time. Very few people will make it to 10 o’clock by this reckoning. An interesting approach might be to look at when sunrise occurred on the day you were born and add the elapsed time to that. For example, the Sun rose at 5:58 on the day I was born and so it’s now about 1:12pm. The Sun set at 5:44pm so I’m not going to make it to sunset. 

Figure 1: clock interface courtesy of https://www.visnos.com/demos/clock

This visualisation may be a way of making a person's diurnal age more comprehensible. When I tell someone that I'm so and so many days old, they are uniformly unimpressed by the statistic. The big takeaway is that one hour approximates to one decade. This is because the number of days in the solar year is very close 360 (either 365 or 366 and averaging about 365.25). Thus when we convert 10 years to days, then treat those days as seconds and convert them to hours, we get 1.014583...

Using a site like timeanddate.com, it's easy to find when the Sun rose on the day you were born. See Figure 2 using my own birth as an example:

Figure 2

Adding 7:14:26 to 5:58:00 (we don't know the exact second for the sunrise), we get 1:12:26 (see Figure 3):

Figure 3: clock interface courtesy of https://www.visnos.com/demos/clock

As I said earlier, I've no chance of reaching sunset but it would certainly be possible for those born at higher latitudes in the winter months. For example, for someone born in London, England, on January 1st 1949, the Sun rose at 8:16 and set at 4:01. Thus the day is a little less than 8 hours long and if you live to 80 then you make it.

This mapping of the days of one's life onto a clock face can produce some previously hidden significances. For example, on what days do the hour and minute hands of the clock overlap? Here is a brief account of what the relevant clock times are (source):

Note that each time they meet, the number of minutes past the hour keeps increasing, so the hour hand would have moved closer to the next number. After 11 o'clock, the minute hand has to travel all the way and by the time they meet it is has to be 12 o'clock again, since we know what the clock looks like at that time. So the two hands overlap 11 times in a 12 hour period. So, in a 24 hour period, they would overlap 22 times.

 To answer the second part of the question, let's try to figure out the little bit of extra time the minute hand needs to catch up to the hour hand after every 1:05 hours. Well, after 12 o'clock there are eleven occasions when the two hands match up, and since the clock hands move at constant speeds, those 11 events are spread equally apart around the clock face, so they are 1/11th of an hour apart. That's 5.454545 minutes apart. In other words they meet after every 1 hour and 5.454545 minutes.

The precise times they overlap (in hours) would be 1 + 1/11, 2 + 2/11, 3+ 3/11, all the way up to 11 + 11/11, which is 12 o'clock again. 

Converting these clock times back to seconds again gives the following sequence:

3927, 7855, 11782, 15709, 19636, 23564, 27491, 31418, 35345, 39273, 43200

I've submitted this sequence for inclusion in the OEIS as doesn't appear there and I think it's an interesting one. Let's see if it gets approved.

UPDATE on August 19th 2020

Today my sequence was approved and with the legendary Neal Sloane giving it the final stamp of approval. It is OEIS A335789. The sequence is a good example of how playing around with different ways of interpreting a number can lead to interesting results. In this case, my viewing the number as representing seconds after 12:00 on a clock face was the crucial step that led me to the devising of the sequence.

A335789

a(n) = time to the nearest second at the n-th instant (n>=0) when the hour and minute hands on a clock face coincide, starting at time 0:00.

Below is a snapshot of the sequence as it appears in the OEIS:

Of course there's nothing new under the sun and following the link OEIS A120500, it can be seen that Lekraj Beedassy, back on August 06th 2006 had exactly the same idea except that the time was represented in HHMMSS format and thus the sequence is self-limiting:

0, 10527, 21055, 31622, 42149, 52716, 63244, 73811, 84338, 94905, 105433, 120000 

A120500

Times in hours, minutes and seconds (to the nearest second) at which the smoothly crossing minute and hour hands of an analog clock coincide, over a period of one complete 12-hour sweep of the hour hand.

Of course, I hadn't seen this sequence when I was developing my own.