## Thursday, November 9, 2017

Happy numbers
$49={4}^{2}+{9}^{2}=16+81=97$
$97={9}^{2}+{7}^{2}=81+49=130$
$130={1}^{2}+{3}^{2}+{0}^{2}=10$
$10={1}^{2}+{0}^{2}=1$

A number is called happy if it leads to 1 after a sequence of steps where in each step number is replaced by sum of squares of its digit that is if we start with Happy Number and keep replacing it with digits square sum, we reach 1; while those that do not end in 1 are unhappy numbers (or sad numbers).
As of 2010, the largest known happy prime is ${2}^{42643801}-1$ (Mersenne prime). Its decimal expansion has 12,837,064 digits.

Pandigital numbers
Pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. In other words, it is a number made up of all distinct digits. The first few pandigital base 10 numbers are given by (sequenceA050278 in the OEIS):
1023456789, 1023456798, 1023456879, 1023456897, 1023456978
A pandigital formula is discovered that generates value of Euler's Number e exceedingly precise.

Narcissistic Numbers

Number that is the sum of its own digits each raised to the power of the number of digits..
$153={1}^{1}+{5}^{5}+{3}^{3}$
35452590104031691935943 is one of the largest Narcissistic prime found.

Keith Number

97 is a Keith number since it generates the sequence:.
1+9+7=17
9+7+17=33
7+17+33=57
17+33+57=107
33+57+107=197

The known prime Keith numbers are 19, 47, 61, 197, 1084051, 74596893730427, ... (OEISA048970). The largest of these is 5752090994058710841670361653731519, which is the largest Keith prime known as of August 2009.

Perfect Number

Number that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, factors of 6 are 1,2,3 & 6.
6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
It is unknown whether there is any odd perfect number, though various results have been obtained.

Munchausen number

Number that is equal to the sum of its digits each raised to the power of itself.
$3435={3}^{3}+{4}^{4}+{3}^{3}+{5}^{5}$
There are only three hexadecimal Munchausen numbers: 1, c4ef722b782c26f, and c76712ffc311e6e.

Sum-product number
Integer that in a given base is equal to the sum of its digits times the product of its digits. for example, 144 is a sum-product number because 1 + 4 + 4 = 9, and 1 × 4 × 4 = 16, and 9 × 16 = 144.
David Wilson have proved that a base 10 sum-product number will not have more than 84 digits.

Friedman Numbers
A Friedman number is an integer, which in a given base, is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, parentheses, and exponentiation. For example, 347 is a Friedman number, since $347={7}^{3}+4$
In a trivial sense, all Roman numerals with more than one symbol are Friedman numbers. VIII = (V - I) × II

Cyclic number
A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142

Taxicab numbers

The smallest number that can be expressed as a sum of two positive cube numbers in ndistinct ways. The most famous taxicab number is 1729 = Ta(2) = 1^3 + 12^3 = 9^3 + 10^3.
The largest taxicab number is

In a given base an integer that is divisible by the sum of its digits when written in that base is Harshad number. Example:
18 is harshad number since 18 is divisible by 9(1+8).
1729 is again an Harshad number as it is divisible by 19(1+7+2+9).

Lychrel number

A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the 196-algorithm. This procedure quickly produces palindromic numbers for most integers. For example, starting with the number 5280 produces palindromic number 23232.
$5280+0825=6105$
$6105+5016=11121$
$11121+12111=23232$
In the 1980s the 196 palindrome problem attracted the attention of microcomputer hobbyists, because 196 is the lowest candidate Lychrel number.