3 Questions on the number 153

[James Magan]

+
M

1.
I believe that the numbers 153, 370, 371 and 407 are the only four "known" numbers which have this property: if you sum the cube of the digits in each number, you get the original number (so in the case of 153, for example, 1 + 125 + 27 = 153).

Can anyone tell me, please, how far mathematical "knowledge" is likely to stretch in this regard: into the thousands? tens of thousands? millions? beyond? Or in other words, what is the "cieling" below which we can say with certainty that there is no other number that has the same property as the four numbers already mentioned?

2.
153 is a triangular number. I would like to access online, or generate on rapidly on an Excel spreadsheet if that is possible, a list of all triangular numbers up to about 1,000,000. I guess that may not be too difficult for a mathematician: could somebody tell me how to do it, please.

3.
3 is a triangular number: it completes the second line of the triangle if you count 1 as being the first line. 3 is also the sum of 1! + 2!.
153 is a triangular number: it completes line 17 of the triangle. 153 is also the sum of 1! + 2! + 3! + 4! + 5!
I would like to be able to view or generate a list of the sums of 1! + 2!; ... 1! + 2! + 3! ; ... 1! + 2! + 3! + 4!; ... 1! + 2! + 3! + 4! + 5!, and so on , increasing by one digit at a time, until the product again reaches about 1,000,000. Could someone tell me how to do that, please.
(I want to know if any other numbers apart from 3 and 153 share the two properties mentioned above.)

Many thanks

 

[Deleted member 4993]

+
M

1.
I believe that the numbers 153, 370, 371 and 407 are the only four "known" numbers which have this property: if you sum the cube of the digits in each number, you get the original number (so in the case of 153, for example, 1 + 125 + 27 = 153).

Can anyone tell me, please, how far mathematical "knowledge" is likely to stretch in this regard: into the thousands? tens of thousands? millions? beyond? Or in other words, what is the "cieling" below which we can say with certainty that there is no other number that has the same property as the four numbers already mentioned?

2.
153 is a triangular number. I would like to access online, or generate on rapidly on an Excel spreadsheet if that is possible, a list of all triangular numbers up to about 1,000,000. I guess that may not be too difficult for a mathematician: could somebody tell me how to do it, please.

3.
3 is a triangular number: it completes the second line of the triangle if you count 1 as being the first line. 3 is also the sum of 1! + 2!.
153 is a triangular number: it completes line 17 of the triangle. 153 is also the sum of 1! + 2! + 3! + 4! + 5!
I would like to be able to view or generate a list of the sums of 1! + 2!; ... 1! + 2! + 3! ; ... 1! + 2! + 3! + 4!; ... 1! + 2! + 3! + 4! + 5!, and so on , increasing by one digit at a time, until the product again reaches about 1,000,000. Could someone tell me how to do that, please.
(I want to know if any other numbers apart from 3 and 153 share the two properties mentioned above.)

Many thanks

[TABLE="width: 453"]
[TR]
[TD]Number[/TD]
[TD]…………………………………...Factorial[/TD]
[TD]……………...Factorial Sum[/TD]
[/TR]
[TR]
[TD="align: right"]1[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]1[/TD]
[/TR]
[TR]
[TD="align: right"]2[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]3[/TD]
[/TR]
[TR]
[TD="align: right"]3[/TD]
[TD="align: right"]6[/TD]
[TD="align: right"]9[/TD]
[/TR]
[TR]
[TD="align: right"]4[/TD]
[TD="align: right"]24[/TD]
[TD="align: right"]33[/TD]
[/TR]
[TR]
[TD="align: right"]5[/TD]
[TD="align: right"]120[/TD]
[TD="align: right"]153[/TD]
[/TR]
[TR]
[TD="align: right"]6[/TD]
[TD="align: right"]720[/TD]
[TD="align: right"]873[/TD]
[/TR]
[TR]
[TD="align: right"]7[/TD]
[TD="align: right"]5040[/TD]
[TD="align: right"]5913[/TD]
[/TR]
[TR]
[TD="align: right"]8[/TD]
[TD="align: right"]40320[/TD]
[TD="align: right"]46233[/TD]
[/TR]
[TR]
[TD="align: right"]9[/TD]
[TD="align: right"]362880[/TD]
[TD="align: right"]409113[/TD]
[/TR]
[TR]
[TD="align: right"]10[/TD]
[TD="align: right"]3628800[/TD]
[TD="align: right"]4037913[/TD]
[/TR]
[TR]
[TD="align: right"]11[/TD]
[TD="align: right"]39916800[/TD]
[TD="align: right"]43954713[/TD]
[/TR]
[TR]
[TD="align: right"]12[/TD]
[TD="align: right"]479001600[/TD]
[TD="align: right"]522956313[/TD]
[/TR]
[TR]
[TD="align: right"]13[/TD]
[TD="align: right"]6227020800[/TD]
[TD="align: right"]6749977113[/TD]
[/TR]
[TR]
[TD="align: right"]14[/TD]
[TD="align: right"]87178291200[/TD]
[TD="align: right"]93928268313[/TD]
[/TR]
[/TABLE]

 

[pka]

.
3 is a triangular number: it completes the second line of the triangle if you count 1 as being the first line. 3 is also the sum of 1! + 2!.
153 is a triangular number: it completes line 17 of the triangle. 153 is also the sum of 1! + 2! + 3! + 4! + 5!
I would like to be able to view or generate a list of the sums of 1! + 2!; ... 1! + 2! + 3! ; ... 1! + 2! + 3! + 4!; ... 1! + 2! + 3! + 4! + 5!, and so on , increasing by one digit at a time, until the product again reaches about 1,000,000. Could someone tell me how to do that, please.
(I want to know if any other numbers apart from 3 and 153 share the two properties mentioned above.)

Triangular numbers.

\(\displaystyle 153 \) is indeed tringular: \(\displaystyle 153=\dfrac{(17)(18)}{2} \)

Generate those numbers by \(\displaystyle \dfrac{(n)(n+1)}{2} \) where \(\displaystyle n\in\mathbb{N} \)(natural numbers).

 

[James Magan]

+
M

Thank you for the various replies, all of which were helpful, and only one of which I did not understand (!).

1. (For Denis) a. I realised after closing down yesterday that a 4-digit number (1999 I thought, but I can see it depends which way you look at it) was going to be the limit for numbers that would be candidates for the select set of those that equal the sum of their cubed digits. Thank you for the confirmation of the other numbers in the set.
(b. "+ over M" is for me the [extremely positive] Universal Equation! I am a Catholic, and it stands for the Cross of Jesus with Mary His Mother standing beneath it. It is based on the coat of arms of the great recent Polish Pope - John Paul II.)

2. (For pka) The way to generate a list of triangular numbers looked wonderful, but it still leaves this dunce (English word ... not sure if it is American) in the dark as to how to do it. If it can be done on an Excel spreadsheet and anyone has got time to do it, could you send it to me? (Up to 200,000,000 would be ideal.) I would be really grateful.

3. (For Subhotosh Khan) Many thanks for the list of the sums of strings of factorials increasing by increments of 1. I had not realised the figures ascended so quickly. Are any of the numbers in that list of 14 Factorial Sums triangular numbers, apart from 3 and 153?

Grateful for your replies. I promise not to ask any more questions for a while.

 

[pka]

2. (For pka) The way to generate a list of triangular numbers looked wonderful, but it still leaves this dunce (English word ... not sure if it is American) in the dark as to how to do it. If it can be done on an Excel spreadsheet and anyone has got time to do it, could you send it to me? (Up to 200,000,000 would be ideal.) I would be really grateful.

Dunce is very much an American word. I am one when it comes to Excel.
But here is a list of the first thousand. On that page you can generate more by changing the n=0 to ?

 

[James Magan]

+
M

Many thanks. I tried n(n+1)/2, n=0 to 5000, but it does not want to do that for me (or at least it is still thinking about it after about half an hour). But I was probably being greedy anyway. My questions are basically answered.

 

[James Magan]

+
M

2 responses:

1.
"Can be done much simpler; just add the natural numbers, like:
0+1 = 1
1+2 = 3
3+3 = 6
6+4 = 10
10+5 = 15
15+6 = 21
and so on..."
- Yes, thank you, I had realised that as I have been pondering low-level triangular numbers for a while now: it seemed to me that a simple programme, such as on a Microsoft Excel document, might be able to generate a string of results if one knew which tools to use. But I think I have got all I need now.

2.
a. "Oh boy...I'm a 110% atheist...not that it matters.
"Hi everyone, my name is Denis and I'm a recovering catholic"...get my drift?

b. Googled "number 153" ; quite "entertaining"..."

Part a: I wish you well; but I do not know where you got the extra ten percent from.
Part b: Yes, 153 is a remarkable number. I do not know much about numbers, but the unique mathematical property of 153, which to me is absolutely mind boggling is the fact that not only does it produce itself if you sum the cube of its digits, but that every number divisible by three in the entire numerical system is reducible to 153 if you sum the cube of its digits, then repeat the process on the product, and keep repeating it ... eventually you will arrive at the irreducible 153!
(So, for example, with the number 3 itself:
3^3 = 27;
2^3 (8) + 7^3 (343) = 351;
3^3(27) + 5^3 (125) + 1^3 (1) = 153.
... but try it on any number that you like that is divisible by 3: stay with something simple if you do not want to go through too many steps to boil it down to 153.)

The reason why this is particularly interesting for a Christian is that 153 is the only exact number of three digits or more which occurs in the four Gospels, which are the core of the New Testament (= that part of the Bible that was written after the life of Christ: hence the part that is proper to Christians).

I do not promise to take this string any further, as it seems already to have got to the existential outskirts of mathematics!