Very special numbers 2

Wagstaff prime: It is a prime number p of the form

where q is another prime. Wagstaff primes are related to the New Mersenne conjecture and have applications in cryptology.

The first three Wagstaff primes are 3, 11, and 43 because

The first few Wagstaff primes are:

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, …

The first exponents q which produce Wagstaff primes or probable primes are:

3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, ……..374321, 986191, 4031399, …

These numbers are proven to be prime for the values of q up to 42737. The largest currently known probable Wagstaff prime

(24031399 + 1)/3

was found by Tony Reix in February 2010. It has 1,213,572 digits and it is the 3rd biggest PRP ever found till this date.

Solinas prime: In mathematics, a Solinas prime, named after Jerome Solinas, is a prime number of the form 2a ± 2b ± 1, where 0 < b < a.

For example, the first five pairs of twin primes are also Solinas primes.

The first few Solinas primes are

3, 5, 7, …

DEVLALI OR SELF NUMBER OR COLOMBIAN NUMBER: In 1963, Kaprekar defined the property which has come to be known as self numbers, which are integers that cannot be generated by taking some other number and adding its own digits to it. For example, 21 is not a self number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer. He also gave a test for verifying this property in any number. These are sometimes referred to as Devlali numbers (after the town where he lived); though this appears to have been his preferred designation, the term self number is more widespread. Sometimes these are also designated Colombian numbers after a later designation.

DEMLO NUMBER: Kaprekar also studied the Demlo numbers, named after a train station where he had the idea of studying them. These are the numbers 1, 121, 12321, …, which are the squares of the repunits* 1, 11, 111, ….

*A repunit is a number consisting of copies of the single digit 1. The term "repunit" was coined by Beiler (1966), who also gave the first tabulation of known factors.

In base-10, repunits have the form:

Rn = (10n – 1)/(10-1) = (10n – 1)/9

Repunits Rn  therefore have exactly  n decimal digits.

Amazingly, the squares of the repunits  (Rn)2 give the Demlo numbers, , , , ...

But, amazingly, this is just the square of the nth repunit (Rn)

 Dn = (Rn)2 (6)

for n ≤ 9, and the squares of the first few repunits are precisely the Demlo numbers: , , , ... . It is therefore natural to find the Demlo umbers as 1, 121, ..., 12345678987654321, 1234567900987654321, 123456790120987654321, ....

The equality (Rn)2 for n ≤ 9 also follows immediately from schoolbook multiplication, as illustrated above. This follows from the algebraic identity

 (7)

The sums of digits of the Demlo numbers for n ≤ 9  are given by

 (8)

More generally, for n = 1, 2, ..., the sums of digits are 1, 4, 9, 16, 25, 36, 49, 64, 81, 82, 85, 90, 97, 106, ... The values of   n for which these are squares are 1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 51, 66, 81, ... corresponding to the Demlo numbers 1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 12345679012345679012345679012345678987654320987654320987654320987654321, ...

SMITH NUMBER:  It is a composite integer with the property that the sum of its digits is the same as the sum of the digits of its prime factors. ...

A composite integer N whose digit sum S(N) is equal to the sum of the digits of its prime factors Sp (N) is called a Smith number.

For example 85 is a Smith number because digit sum of 85 i.e. S(85) = 8 + 5=13, which is equal to the sum of the digits of its prime factors i.e. Sp (85) = Sp (17 x 5) = 1 + 7 + 5 = 13.

While skimming his phone directory in 1982, Albert Wilansky, a mathematician of Lehigh University , noticed that the telephone number of his brother-in-law H. Smith had the following peculiar property: The sum of the digits of that number was equal to the sum of the digits of the prime factors of that number. Smith's telephone number was 493-7775. This number can be written as the product of its prime factors in the following way:

The sum of all digits of the telephone number is 4+9+3+7+7+7+5=42, and the sum of the digits of its prime factors is equally 3+5+5+6+5+8+3+7=42. Wilansky was so amazed by his discovery that he named this type of numbers after his brother-in-law: Smith numbers.

Wilansky also mentioned two other numbers with this property i.e. 9985 and 6036.  It is now known that there are infinitely many Smith numbers.

Jens Kruse Andersen reported on 28th April 2008 that there are 2335807857 smith numbers below 1011.

The Beast number 666 is also a Smith Number.

There are 376 Smith numbers below 10000. These are: 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690,…..

Palindromic Smith numbers:  Palindromic Smith numbers below 106 are:

4, 22, 121, 202, 454, 535, 636, 666, 1111, 1881, 3663, 7227, 7447, 9229, 10201, 17271, 22522, 24142, 28182, 33633, 38283, 45054, 45454, 46664, 47074, 50305, ………., 717717, 832238, 841148, 864468, 951159, 956659, 974479 and 983389.

Reversible Smith Numbers: Smith numbers, whose reversal is also a smith number, can be termed as Reversible Smith numbers.

Reversible Smith numbers below 104 are:

4, 22, 58, 85, 121, 202, 265, 319, 454, 535, 562, 636, 666, 913, 1111, 1507, 1642, 1881, 1894, 1903, 2461, 2583, 2605, 2614, 2839, 3091, 3663, 3852, 4162, 4198, 4369, ………, 6439, 6835, 7051, 7227, 7249, 7438, 7447, 8158, 8185, 8347, 8518, 8545, 8874, 8914, 9229, 9346, 9355, 9382, 9427 and 9634.

All palindromic smith numbers mentioned above are special cases of reversible smith numbers.

NARCISSISTIC NUMBER: It is a number which is equal to the sum of the cubes of its digits:

e.g. 153  =  13  +  53  +  33  =  1   +  125 +  27  =  153

Also note that: 153 = 1 + 2 + 3 + 4 + ... + 16 + 17, or 153 is the 17th triangular number.

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

PERFECT DIGITAL INVARIANT NUMBER: It is a number that can be expressed as some combination of its digits and various operations, powers etc.

`e.g.1634  =  14  +  64  +  34  +  44   =            1   + 1296 +  81  + 256  =  1634`

The two numbers discussed so far used the cubes and 4th powers of the digits. Numbers that use higher powers also exist; some examples are:

`                      54,748 and fifth powers,`
`              548,834 and sixth powers,`
`            1,741,725 and seventh powers.`
` `

RAMANUJAM NUMBER OR HARDY-RAMANUJAM NUMBER: This number 1729 is the smallest number which can be expressed as the sum of two positive cubes in two different ways.

1729 = 123 + 13 = 93 + 103

It is also a Niven number, that is, a number divisible by the sum of its digits.

1729 = 19 x 91; we see that its factors are palindromes of each other.

CAROL NUMBER: A Carol number is an integer of the form 4n − 2n + 1 − 1. An equivalent formula is (2n − 1)2 − 2. The first few Carol numbers are: −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527

Starting with 7, every third Carol number is a multiple of 7. Thus, for a Carol number to also be a prime number, its index n cannot be of the form 3x + 2 for x > 0. The first few Carol numbers that are also prime are 7, 47, 223, 3967, 16127.  As of July 2007[update], the largest known Carol number that is also a prime is the Carol number for n = 253987, which has 152916 digits.

The 7th Carol number and 5th Carol prime, 16127, is also a prime when its digits are reversed, so it is the smallest Carol emirp. The 12th Carol number and 7th Carol prime, 16769023, is also a Carol emirp.

PELL NUMBER: In mathematics, the Pell numbers are an infinite sequence of integers that have been known since ancient times, the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell-Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.

STIRLING NUMBER: Stirling numbers of the first kind:

Unsigned Stirling numbers of the first kind

(with a lower-case "s") count the number of permutations of n elements with k disjoint cycles.

Stirling numbers of the first kind (without the qualifying adjective unsigned) are the coefficients in the expansion

where (x)n is the Pochhammer symbol for the falling factorial,

Note that (x)0 = 1 because it is an empty product. Combinatorialists also sometimes use the notation for the falling factorial, and for the rising factorial.

Stirling numbers of the second kind:

Stirling numbers of the second kind count the number of ways to partition a set of n elements into k nonempty subsets. The sum

is the nth Bell number. Using falling factorials, we can also characterize the Stirling numbers of the second kind by

The Lah numbers, are sometimes being referred as Stirling numbers of the third kind.

LAH NUMBER: In mathematics, Lah numbers, in 1955, are coefficients expressing rising factorials in terms of falling factorials.

Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets. Lah numbers are related to Stirling numbers.

Unsigned Lah numbers:

Signed Lah numbers:

L(n, 1) is always n!; using the interpretation above, the only partition of {1, 2, 3} into 1 set can be ordered in 6 ways:

{(1, 2, 3)}, {(1, 3, 2)}, {(2, 1, 3)}, {(2, 3, 1)}, {(3, 1, 2)} or {(3, 2, 1)}

L(3, 2) corresponds to the 6 partitions with two ordered parts:

{(1), (2, 3)}, {(1), (3, 2)}, {(2), (1, 3)}, {(2), (3, 1)}, {(3), (1, 2)} or {(3), (2, 1)}

L(n, n) is always 1: partitioning {1, 2, 3} into 3 non-empty subsets results in subsets of length 1.

{(1), (2), (3)}

Paraphrasing Karamata-Knuth notation for Stirling numbers, it was proposed to use the following alternative notation for Lah numbers:

FRIEDMAN NUMBER: 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 (+, −, ×, ÷) and sometimes exponentiation. For example, 347 is a Friedman number since 347 = 73 + 4. The first few base 10 Friedman numbers are:

25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024, 1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, 3125, 3159.

Parentheses can be used in the expressions, but only to override the default operator precedence, for example, in 1024 = (4 − 2)10. Allowing parentheses without operators would result in trivial Friedman numbers such as 24 = (24). Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 001729 = 1700 + 29.

A nice Friedman number is a Friedman number where the digits in the  expression can be arranged to be in the same order as in the number itself. For example, we can arrange 127 = 27 − 1 as 127 = −1 + 27. The first nice Friedman numbers are:

127, 343, 736, 1285, 2187, 2502, 2592, 2737, 3125, 3685, 3864, 3972, 4096, 6455, 11264, ……., 15667, 15688, 16377, 16384, 16447, 16875, 17536, 18432, 19453, 19683, 19739..

SOME SPECIAL NUMBERS:

We now list some more special numbers:

1.The number 635,318,657 is one such, for:

635,318,657 = A4 + B4 = C4 + D4

where A, B, C, and D are distinct whole numbers, to be found from the clues given below:

Clue #1: One pair of numbers are consecutive, in the 130-140 range.

Clue #2: One number of the second pair is 59.

`2.The number 145 = 145 =  1!  +  4!  +  5!   =  1   +  24  +  120=  145`
`3.The number 142857:         `
`           142857 × 2  =  285714`
`                      142857 × 3  =  428571`
`                      142857 × 4  =  571428`
`                      142857 × 5  =  714285`
`                      142857 × 6  =  857142`
`                          142857 x  7 =   999999`

Also,  142 + 857 = 999

And 14 + 28 + 57 = 99

Finally, let's square the number before splitting and adding.

1428572 = 20,408,122,449 = 20408 + 122449.

4.The number 1992:

Also,   1992 = 8 × 3 × 83

SOME MORE SPECIAL NUMBERS: Consider a two-digit number, say, ab, which is 10 a + b in decimal notation.

Suppose na + b + ab = 10a + b which is the original number, where n is a non-zero number less than or equal to 9. What would the numbers look like?

The above implies b= 10 – n

We see some examples now:

n = 1;  The numbers are 19, 29, 39, 49, 59, 69, 79, 89, 99.

n = 2; The numbers are 18, 28, 38, 48, 58, 68. 78, 88, 98, and so on.

was found by Tony Reix in February 2010. It has 1,213,572 digits and it is the 3rd biggest PRP ever found at this date.

.

.

:

SOME MORE SPECIAL NUMBERS:

153 = 13 + 53 + 33 = 1! + 2! + 3! + 4! + 5!

1634 = 14 + 64 + 34 + 44 = 1 + 1296 + 81 + 256

1729 = 19 x 91 (Product of two palindromic numbers) = 7 x 13 x 19 (Product of three primes in A.P.)

145 = 1! + 4! + 5!

A mnemonic for Pi: How I wish I could recollect Pi,

Eureka cried the great inventor,

Christmas pudding, Christmas pie,

Is the problem’s very centre.

BELL NUMBER: In combinatorics, the nth Bell number, named after Eric Temple Bell, is the number of partitions of a set with n members, or equivalently, the number of equivalence relations on it. Starting with B0 = B1 = 1, the first few Bell numbers are:

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, …

HAPPY NUMBER: A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers)

If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of its sequence are unhappy.

For example, 7 is happy, as the associated sequence is:

72 = 49

42 + 92 = 97

92 + 72 = 130

12 + 32 + 02 = 10

12 + 02 = 1.

The happy numbers below 500 are:

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, ……397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496

The happiness of a number is preserved by rearranging the digits, and by inserting or removing any number of zeros anywhere in the number.

The unique combinations of above (the rest are just rearrangements and/or insertions of zero digits):

1, 7, 13, 19, 23, 28, 44, 49, 68, 79, 129, 133, 139, 167, 188, 226, 236, 239, ….., 556, 566, 888, 899

If n is not happy, then its sequence does not go to 1. What happens instead is that it ends up in the cycle

4, 16, 37, 58, 89, 145, 42, 20, 4, ...

PERRIN NUMBER: Perrin numbers are a sequence of numbers similar to the Fibonacci numbers in that successive numbers are derived from two previous adjacent numbers, only in this case we skip a place so that each number can be obtained from the sum of the numbers two and three places previous.

Expressed more clearly as a recurrence relation the Perrin numbers are defined as:

P(n): P(0) = 3, P(1) = 0, P(2) = 3, and P(n) = P(n-2) + P(n-3)

So that the first handful of Perrin numbers would be:

3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119, 158, etc...

CULLEN NUMBER: In mathematics, a Cullen number is a natural number of the form n · 2n + 1 (written Cn). Cullen numbers are special cases of Proth numbers.

In that sense, almost all Cullen numbers are composite. Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2n+a + b where a and b are integers, and in particular also for Woodall numbers. The only known Cullen primes are those for n equal:

1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881

Still, it is conjectured that there are infinitely many Cullen primes.

In 1976 Christopher Hooley showed that the natural density of positive integers  for which Cn is a prime is of the order o(x) for .

Still, it is conjectured that there are infinitely many Cullen primes.

As of August 2009[update], the largest known Cullen prime is 6679881 × 26679881 + 1. It is a megaprime with 2,010,852 digits and was discovered by a Prime Grid participant from Japan.

Sometimes, a generalized Cullen number is defined to be a number of the form n · bn + 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Cullen prime. Woodall numbers are sometimes called Cullen numbers of the second kind.

PROTH NUMBER: A Proth number is a number of the form for odd , a positive integer, and . The condition is needed since otherwise, every odd number would be a Proth number. The first few Proth numbers are 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, ...

The Cullen numbers are a special case of the Proth numbers with k = n (and the inequality restriction dropped). The Fermat numbers are a special case of the Proth numbers with  k = 1.

WOODALL NUMBER: In number theory, a Woodall number (Wn) is any natural number of the form

Wn = n × 2n − 1

for some natural number n. The first few Woodall numbers are:

1, 7, 23, 63, 159, 383, 895, …

Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers Wn are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, …; the Woodall primes themselves begin with 7, 23, 383, 32212254719, … It is conjectured that there are infinitely many Woodall primes. As of December 2007[update], the largest known Woodall prime is 3752948 × 23752948 − 1.  It has 1,129,757 digits and was found by Matthew J. Thompson in 2007.

CUNNINGHAM NUMBER: Cunningham numbers are a simple type of binomial number, they are of the form

where b and n are integers and b is not already a power of some other number. They are denoted C±(bn).

“Homogeneous Cunninghman numbers" is the term used for those numbers of the form an ± bn which seem not otherwise to have a name. Cunningham numbers proper take the form an ± 1, where a is an integer, not a prime power, between 2 and 12 inclusive.

A joke: PIZZA = Pi Z2 A = Volume of a right circular cylinder with radius z and thickness ‘A’.