CURIOUS PROPERTIES OF 153 |
References about number 153 can be found in the New Testament, where in the net Simon Peter drew from the Sea of Tiberias held 153 fishes. Some of the interesting properties of number 153 published in [1] are:
Curious properties of number 153:
153 = 1^{3 }+ 5^{3 }+ 3^{3}
153 = 1! + 2! + 3! + 4! + 5!
1 + 5 + 3 = 9 = 3^{2}
1 + 3 + 9 + 17 + 51 = 81 = 9^{2}
Aliquot divisors of a number are all the divisors of that number excluding the number itself but including 1. It is seen that the sum of aliquot divisors of 153 is the square of the sum of the digits of 153.
153 + 351 = 504
504^{2} = 288 x 882
153 / (1 + 5 + 3) = 17
Since reverse of 153, i.e. 351 is also a Harshad number(or Niven Number), 153 can be termed as a reversible Harshad number(or reversible Niven Number).
153 = 3 * 51
Note that the digits used in multipliers are same as in product.
135 = 1^{1} + 3^{2} + 5^{3}
1 + 3 + 9 + 17 + 51 + 153 = 234
The product of aliquot divisors of 153 is 23409:
1 * 3 * 9 * 17 * 51 = 23409
Note that the product of aliquot divisors of 153 contain the sum of all divisors of 153 juxtaposed:
23409 = 234:09
234 = Sum of all divisors of 153
09 = Square root of the sum of aliquot divisors of 153.
For example: Take the number 108
1^{3} + 0^{3} + 8^{3} = 513 and 5^{3} + 1^{3 }+ 3^{3} = 153
So, the number 108 reaches 153 in two cycles and it can be represented as
108→513→153
A detailed study of all numbers up to 10^{5}reveals that all numbers which are multiple of 3 and are less than 10^{5} reach 153 (after the repeated process of summing the cubes of digits is done) in maximum 14 cycles. However, maximum 13 cycles is required for all numbers, which are multiple of 3 and are less than 10,000.
The smallest number, which requires 13 cycles to reach 153, is 177, i.e.,
177→ 687→ 1071→345→ 216→ 225→ 141→ 66→ 432→ 99→ 1458→ 702→ 351→ 153
Table 1 indicates the smallest numbers, which reach 153 in cycles from 1 to 14.
TABLE: 1
No. of cycles |
Smallest number |
1 |
135 |
2 |
18 |
3 |
3 |
4 |
9 |
5 |
12 |
6 |
33 |
7 |
114 |
8 |
78 |
9 |
126 |
10 |
6 |
11 |
117 |
12 |
669 |
13 |
177 |
14 |
12558 |
The Smallest Number (Of course a multiple of 3) which reaches 153 in 15 cycles is 44499999999999999999, which in short can be represented as 4_{3}9_{17} i.e.
44499999999999999999® 12585® 771®687 ® 1071® 345®216® 225® 141® 66® 432® 99® 1458® 702® 351® 153
Jörg Zurkirchen vide his email dated 16th aug 2021 advised that Jon E. Schoenfield found the smallest numbers, which reach 153 in 16 and 17 cycles (Sloane's A346630).
a(16) = 3.777999...999*10^61042524005486970; it has one 3, three 7's, and 61042524005486967 9's, so the sum of the cubes of its digits is 1*3^{3} + 3*7^{3} + 61042524005486967*9^{3} = 44499999999999999999 = a(15).
a(17) consists of the digit string 45888 followed by a very, very long string of 9's. The number of 9's in that string is (a(16) - 1725)/729,
which is a 61042524005486968-digit number consisting of the digit 5 followed by 753611407475147 copies of the 81-digit string
182441700960219478737997256515775034293552812071330589849108367626886145404663923 followed by a single instance of the 60-digit string
182441700960219478737997256515775034293552812071330589849106
He also shows how to find additional numbers (Sloane's A346789).
Some New Observations on number 153:
p (153) = p (15) * 3!
Let us say 153 increasingly from left to right:
1, 15, 153
We find that 115153 is prime?(Sent by G.L.Honaker,Jr. by email dt 3rd Feb, 2002)
T_{n} * 1225 + 153 is a triangular number.
For example:
1*1225 + 153 = 1378 = T_{52}
3*1225 + 153 = 3828 = T_{87}
6*1225 + 153 = 7503 = T_{122}
10*1225 + 153 = 12403 = T_{157}
A less described feature of 153 emerges if the number is read as a binary number: 10011001
10011001 when viewed linear is a palindrome comprising 8 positions. If 10011001 is circularized,
forming an octagonal ring,
more patterns emerge.
Note that the maximal number in a byte (255) is included in above figure.
The circularized 10011001 sequence gives a multiple of 51 (or 3 x 17) from any position the sequence is read.
(Sent by Stefan Krauss, University of Oslo by email dated 29 August 2017)
If you find any new and interesting observation about 153, please email me.
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[1] Curious Properties of 153, Shyam Sunder Gupta, Science Reporter, February 1991, India.