Mathematically 244

1. 244 is an anti-perfect. 

  Its proper divisors are 1, 2, 4, 61, and 122, & adding their reversal is 244.

                           1 + 2 + 4 + 16 + 221 = 244.

2. It is a palindromic in bases 3 (100001₃), 11 (202₁₁), 60 (44₆₀), 121 (22₁₂₁), 243 (11₂₄₃).

3. 244 is also the sum of three cubes,
                                   244=1^3+3^3+6^3 

4. It is the sum of two nonzero fifth powers
                                  244 = 1^5 + 3^5

5. It is a Harshad number on the base 3, 9, 11 ... . Harshad number was discovered by Indian mathematician Kaprekar.

Dr Rajesh Kumar Thakur

Mathematically 240

240

1. It is a semiperfect number.

2. It is a highly composite number since it has 20 divisors total (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, and 240), more than any previous number.

3. It is a refactorable number or tau number, since it has 20 divisors and 20 divides 240.

4. It is a pronic number since it is the product of 2 consecutive number
240 = 15 × 16

5. 240 can be expressed as a sum of consecutive primes in two different ways: 240 = 53 + 59 + 61 + 67
 = 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43.

6. It is the product of 6 Fibonacci number.
240= 1 × 1 × 2 × 3× 5× 8

Number 211

Number 211

30th July in a normal year and 31st July in a leap year is the 211th day of the year
1. It is an odd number

2. It is the sum of three consecutive primes.
211 = 67 + 71 + 73

3. It is a Chen Prime. A prime number p is called a Chen prime if p + 2 is either a prime or product of two primes.

Rajesh Kumar Thakur

Number 210

Number 210

1.  (21, 20, 29) and (35, 12, 37) are the two least primitive Pythagorean triangles with different hypotenuses and the same area (=210).

2. It is the product of first four prime number
                         210 = 2 x 3 x 5 x 7

3. It is the sum of 8 consecutive prime numbers.
                   13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210

4. It is a triangular Number. 
                   190              210                    231

5. It is a Harshad Number. To know more about Harshad Number. Harshad number was discovered by Indian Mathematician Kapekar.

6. It is a Pronic Number. Pronic number is a number which is the product of n and n+ 1
                210 = 14 x 15


7. It is an abundant Number. Abundant number is a number whose sum of proper divisor is more than the number. 
12 is the first abundant number.
Factor of 12 are 1, 2, 3, 4 and 6
1 + 2 + 3 + 4 + 6 > 12

8. It is a composite number, even number and pentagonal number.

Dr Rajesh Kumar Thakur

Brahmagupta - Fibonacci Identity

                                                      Brahmagupta - Fibonacci Identity 

There is a very famous identity used in algebra known as – Brahmagupta – Fibonacci identity that expresses the product of two sums of two squares as a sum of two squares in two different ways.

                        (a² + b²) (c² + d²) = (ac – bd)² + (ad + bc)²      

                                                  = ( ac + bd)² + (ad – bc)²

For example:-

(1² + 4²) (2² + 7²) = (2 – 28)² + (7 + 8)²

                            = 26² + 15²

And

(1² + 4²) (2² + 7²) = (2 + 28)² + (7 – 8)²

                            = 30² + 1²

^{Dr Rajesh Kumar Thakur}

Mathematically 101

                                                               101

1. One hundred one is a sexy prime as well as a sexy prime triplet. Sexy primes differ from each other by 6. If p is a prime and p + 6 is also a prime then it is called a sexy prime.
Example:- (5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107)

2. 101 is also a sexy prime triplet. If p, p+6 and p + 12 are all prime then it is called sexy prime triplet.
Example:- (5,11,17), (7,13,19), (17,23,29), (31,37,43), (47,53,59), (67,73,79), (97,103,109), (101,107,113),

3. It is the sum of 5 consecutive primes
                                       13 + 17 + 19 + 23 + 29 

4. It can be written as the sum of first 5 factorial with alternate sign.
                                  101 = 5! - 4! + 3! - 2! + 1!

5. It is a palindrome number and a palindrome prime.

6. It is the largest prime in form of 10^n + 1
                                   101 = 10^2 + 1

7. There are 101 digits in the product of the 39 successive primes produced by the formula n² + n + 41, where n = 1 to 39. This formula was used by Charles Babbage to demonstrate the capabilities of his Difference Engine (1819-1822))

Happy reading 
Dr Rajesh Kumar Thakur

Self Number

                                                          D R Kaprekar (1905 -1986)
Self number / Devlali Number / Swayambhu' Number was discovered by Indian mathematician Dattathreya Ramchandra Kaprekar who was born in Dhanu (Maharastra). 

A self number is a number that cannot be written as the sum of any other integer n and the individual digits of n. It was discovered in 1949 by Kaprekar. 

These are the self number 
1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244 - - -

Start with a number, say 23. The sum of its digits (2 + 3) are 5 which we add to 23 to obtain 28. Again add 2 and 8 to get 10 which we add to 28 to get 38. Continuing gives the sequence

23, 28, 38, 49, 62, 70, ...

These are all generated by 23. But is 23 generated by a smaller number? Yes, 16 generates 23. In fact the sequence we looked at really starts at 1

1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, ... 

Here is the recurrence formula to find the Self Number

where 

(Sources :- Wikipedia / Mac Tutor Archive )

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rkthakur1974@gmail.com

Dr Rajesh Kumar Thakur