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 = 13 + 53 + 33

153 = 1! + 2! + 3! + 4! + 5!

1 + 5 + 3 = 9 = 32

1 + 3 + 9 + 17 + 51 = 81 = 92

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

5042 = 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 = 11 + 32 + 53

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

13 + 03 + 83 = 513 and 53 + 13 + 33 = 153

So, the number 108 reaches 153 in two cycles and it can be represented as

108513153

A detailed study of all numbers up to 105reveals that all numbers which are multiple of 3 and are less than 105 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

 

It may be noted that the smallest number to reach 153 in 15 cycles will be more than 1019. Can you find it?

 

Some New Observations on number 153:

p (153) = p (15) * 3!

 

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.

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