Amicable Numbers With Patterns in Products and Powers

There are many ways of writing amicable numbers. One with divisions and sums. The other with pair of powers of each other. There is another way to represent is in product. In this paper, we brings amicable numbers in pairs in terms of products and powers. The idea of self-amicable is also introduced. Few blocks of symmetrical amicable numbers multiples of 10 are also given. Some interesting patterns among amicable numbers are also given

Triangular-Type Selfie Numbers

Numbers represented by their own digits by certain operations are considered as selfie numbers. There are many ways of writing selfie numbers, such as, numbers written in digit's order or its reverse just with basic operations. We can extend them by use of other operations, such as, factorial, square-root, Triangular numbers, Fibonacci sequence values, etc. In this work, the selfie numbers are written by use of triangular numbers in digit's order and reverse

Fibonacci Sequence Type Selfie Numbers: Basic Operations

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci sequence, Triangular numbers, etc. These operations are applied for single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factorial and square-root. This paper extends authors previous work for Fibonacci sequence type selfie numbers in basic operations. For the operations, such as, factorial and square-root, the work shall be given elsewhere. The work is in digit's order and in reverse order of digits, and is up to 5-digits numbers. This extends considerably, author's previous work

Different Types of Magic Squares of Orders 6, 8, 10 and 12

<p>It is revised version of author's previous work. It brings new ideas of construction of magic squares. This work is for magic squares of even orders 6, 8, 10 and 12. The ideas ideas used to bring these magic square are: <strong>bordered magic rectangles, bordered double digits magic rectangles, cornered magic rectangles, striped magic rectangles</strong>, etc. When the length and width are equal these magic rectangles becomes as magic squares.  Another idea used is of algebraic formula <strong>(a+b)^2</strong>. Here we consider small blocks of magic squares and magic rectangles, such as <strong>a^2</strong>, <strong>b^2</strong>, <strong>axb</strong> and <strong>bxa.</strong> </p&gt

Striped Magic Squares of Even Orders 6, 8, 10, 12 and 14

<p>This work brings <strong>striped magic squares</strong> of even orders 6, 8, 10, 12 and 14> These are constructed based on <strong>equal width magic rectangles</strong>. <strong>Equal width magic rectangles</strong> are of type 2x4, 2x6, 2x8, etc. Magic squares constructed based on <strong>equal width magic rectangles</strong>, we call as <strong>striped magic squares.</strong></p&gt

Striped Magic Squares of Even Orders 6, 8, 10 and 12

<p>This work brings magic squares of even orders 6, 8, 10 and 12 based on <strong>equal width magic rectangles</strong>. <strong>Equal width magic rectangles</strong> are of type 2x4, 2x6, 2x8, etc. Magic squares constructed based on <strong>equal width magic rectangles</strong>, we call as <strong>striped magic squares.</strong></p&gt

Different Types of Magic Squares of Orders 6, 8, 10 and 12

<p>It is revised version of author's previous work. It brings new ideas of construction of magic squares. This work is for magic squares of even orders 6, 8, 10 and 12. The ideas ideas used to bring these magic square are: <strong>bordered magic rectangles, bordered double digits magic rectangles, cornered magic rectangles, striped magic rectangles</strong>, etc. When the length and width are equal these magic rectangles becomes as magic squares.  Another idea used is of algebraic formula <strong>(a+b)^2</strong>. Here we consider small blocks of magic squares and magic rectangles, such as <strong>a^2</strong>, <strong>b^2</strong>, <strong>axb</strong> and <strong>bxa.</strong> We are able to bring 6 magic squares of order 6, 30 magic squares of order 8, 175 magic squares of order 10 and 634 magic squares of order 12. These are available in author's site, whose links are given above.</p&gt

Nested Inequalities Among Divergence Measures

In this paper we have considered an inequality having 11 divergence measures. Out of them three are logarithmic such as Jeffryes-Kullback-Leiber [4] [5] J-divergence. Burbea-Rao [1] Jensen-Shannon divergence and Taneja [7] arithmetic-geometric mean divergence. The other three are non-logarithmic such as Hellinger discrimination, symmetric A2?divergence, and triangular discrimination. Three more are considered are due to mean divergences. Pranesh and Johnson [6] and Jain and Srivastava [3] studied different kind of divergence measures.We have considered measures arising due to differences of single inequality having 11 divergence measures in terms of a sequence. Based on these differences we have obtained many inequalities. These inequalities are kept as nested or sequential forms. Some reverse inequalities and equivalent versions are also studied

Representations of Letters and Numbers With Equal Sums Magic Squares of Orders 4 and 6

This work brings 26 letters from A to Z and 10 numbers from 0 to 9 in terms of blocks of magic squares of orders 4 and 6. Letters and numbers ares constructed with blocks of equal sums magic squares of orders 4 and 6. In each case, consecutive natural numbers are used starting from 1, and there is no repetition of number