Some special numbers

Jai Ganesh · 2024-06-06 23:25:46

62) Eddington Number

In astrophysics, the Eddington number,

, is the number of protons in the observable universe. Eddington originally calculated it as about ; current estimates make it approximately

The term is named for British astrophysicist Arthur Eddington, who in 1940 was the first to propose a value of

and to explain why this number might be important for physical cosmology and the foundations of physics.

History

Eddington argued that the value of the fine-structure constant, α, could be obtained by pure deduction. He related α to the Eddington number, which was his estimate of the number of protons in the universe. This led him in 1929 to conjecture that α was exactly 1/136. He devised a "proof" that

, or about . Other physicists did not adopt this conjecture and did not accept his argument.

In the late 1930s, the best experimental value of the fine-structure constant, α, was approximately 1/137. Eddington then argued, from aesthetic and numerological considerations, that α should be exactly 1/137.

Current estimates of

point to a value of about . These estimates assume that all matter can be taken to be hydrogen and require assumed values for the number and size of galaxies and stars in the universe.

During a course of lectures that he delivered in 1938 as Tarner Lecturer at Trinity College, Cambridge, Eddington averred that:

I believe there are 15747724136275002577605653961181555468044717914527116709366231425076185631031296 protons in the universe and the same number of electrons.

This large number was soon named the "Eddington number".

Shortly thereafter, improved measurements of α yielded values closer to 1/137, whereupon Eddington changed his "proof" to show that α had to be exactly 1/137.

Jai Ganesh · 2024-06-26 23:11:53

63) Number 4

Jai Ganesh · 2024-06-27 20:32:13

64) Powers

Jai Ganesh · 2024-06-27 20:52:02

65)

Jai Ganesh · 2024-07-07 17:41:48

66) 17

.

Jai Ganesh · 2024-07-07 18:06:25

67) Weird number

In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.

Examples

The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but not weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2 + 4 + 6 = 12.

The first few weird numbers are

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ...

Properties

Infinitely many weird numbers exist. For example, 70p is weird for all primes p ≥ 149. In fact, the set of weird numbers has positive asymptotic density.

It is not known if any odd weird numbers exist. If so, they must be greater than

Jai Ganesh · 2024-07-08 19:51:31

68) Achilles number

An Achilles number is a number that is powerful but not a perfect power. A positive integer n is a powerful number if, for every prime factor p of n,

is also a divisor. In other words, every prime factor appears at least squared in the factorization. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as , where m and k are positive integers greater than 1.

Achilles numbers were named by Henry Bottomley after Achilles, a hero of the Trojan war, who was also powerful but imperfect. Strong Achilles numbers are Achilles numbers whose Euler totients are also Achilles numbers; the smallest are 500 and 864.

The Achilles numbers up to 5000 are:

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000.

The smallest pair of consecutive Achilles numbers is:

Examples

As an example, 108 is a powerful number. Its prime factorization is

, and thus its prime factors are 2 and 3. Both are divisors of 108. However, 108 cannot be represented as , where m and k are positive integers greater than 1, so 108 is an Achilles number.

The integer 360 is not an Achilles number because it is not powerful. One of its prime factors is 5 but 360 is not divisible by

Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also

are divisors of it. It is a perfect power:

So it is not an Achilles number.

The integer

is a strong Achilles number as its Euler totient of
is also an Achilles number.

Jai Ganesh · 2024-07-14 21:35:47

69) 255

In mathematics

Its factorization makes it a sphenic number. Since

it is a Mersenne number (though not a pernicious one), and the fourth such number not to be a prime number. It is a perfect totient number, the smallest such number to be neither a power of three nor thrice a prime.

Since 255 is the product of the first three Fermat primes, the regular 255-gon is constructible.

In base 10, it is a self number.

255 is a repdigit in base 2 (11111111), in base 4 (3333), and in base 16 (FF).

In computing

255 is a special number in some tasks having to do with computing. This is the maximum value representable by an eight-digit binary number, and therefore the maximum representable by an unsigned 8-bit byte (the most common size of byte, also called an octet), the smallest common variable size used in high level programming languages (bit being smaller, but rarely used for value storage). The range is 0 to 255, which is 256 total values.

For example, 255 is the maximum value of

* components in the 24-bit RGB color model, since each color channel is allotted eight bits;
* any dotted quad in an IPv4 address; and
* the alpha blending scale in Delphi (255 being 100% visible and 0 being fully transparent).

This number could be interpreted by a computer as -1 if a programmer is not careful about which 8-bit values are signed and unsigned, and the two's complement representation of -1 in a signed byte is equal to that of 255 in an unsigned byte.

Jai Ganesh · 2024-07-15 23:09:25

70)

Jai Ganesh · 2024-07-17 00:49:45

71) The Third and Fourth Perfect numbers

The first and second perfect numbers are 6 and 28 respectively.

The Third Perfect Number

496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime

yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case.

The Fourth Perfect Number

It is most notable for being a perfect number (its divisors 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, and 4064 add up to 8128), and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime

yielding 8128. Also related to its being a perfect number, 8128 is a harmonic divisor number

Jai Ganesh · 2024-07-17 21:18:46

72) Cyclic Number

A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are

Relation to repeating decimals

Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of

1/(L + 1).
Conversely, if the digital period of 1/p (where p is prime) is

p - 1,
then the digits represent a cyclic number.

Form of cyclic numbers

From the relation to unit fractions, it can be shown that cyclic numbers are of the form of the Fermat quotient

where b is the number base (10 for decimal), and p is a prime that does not divide b. (Primes p that give cyclic numbers in base b are called full reptend primes or long primes in base b).

For example, the case b = 10, p = 7 gives the cyclic number 142857, and the case b = 12, p = 5 gives the cyclic number 2497.

Not all values of p will yield a cyclic number using this formula; for example, the case b = 10, p = 13 gives 076923076923, and the case b = 12, p = 19 gives 076B45076B45076B45. These failed cases will always contain a repetition of digits (possibly several).

The first values of p for which this formula produces cyclic numbers in decimal (b = 10) are :

7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823, 857, 863, 887, 937, 941, 953, 971, 977, 983, ...

Properties of cyclic numbers

* When multiplied by their generating prime, the result is a sequence of b - 1 digits, where b is the base (e.g. 10 in decimal). For example, in decimal,

* When split into groups of equal length (of two, three, four, etc... digits), and the groups are added, the result is a sequence of b - 1 digits. For example, 14 + 28 + 57 = 99, 142 + 857 = 999, 1428 + 5714+ 2857 = 9999, etc. ... This is a special case of Midy's Theorem.
* All cyclic numbers are divisible by b - 1 where b is the base (e.g. 9 in decimal) and the sum of the remainder is a multiple of the divisor.

It can be shown that no cyclic numbers (other than trivial single digits, i.e. p = 2) exist in any numeric base which is a perfect square, that is, base 4, 9, 16, 25, etc.

142857

The number 142,857 is a Kaprekar number. 142857, the six repeating digits of 1/7 (0.142857), is the best-known cyclic number in base 10. If it is multiplied by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of
1/7, 2/7, 3/7, 4/7, 5/7, or 6/7 respectively.

It is the repeating part in the decimal expansion of the rational number 1/7 = 0.142857. Thus, multiples of
1/7 are simply repeated copies of the corresponding multiples of 142857.

Jai Ganesh · 2024-07-17 23:47:24

73) The Number 65536

65536 is

, so in tetration notation 65536 is .

When expressed using Knuth's up-arrow notation, 65536 is

, which is equal to

, which is equivalent to

or

As

is also equal to 4, or

can thus be written as

, or
or as the pentation,
(hyperoperation notation).

65536 is a superperfect number - a number such that

.

Jai Ganesh · 2024-07-18 21:38:15

74) Pandigital number

In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 (one billion two hundred thirty-four million five hundred sixty-seven thousand eight hundred ninety) is a pandigital number in base 10. The first few pandigital base 10 numbers are given by :

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689 ...

The smallest pandigital number in a given base b is an integer of the form

.

Jai Ganesh · 2024-07-19 20:25:46

75)

.

Jai Ganesh · 2024-07-19 22:10:55

76)

Jai Ganesh · 2024-07-20 01:03:01

77)

Jai Ganesh · 2024-07-20 13:40:13

78)

.

Jai Ganesh · 2024-07-21 00:03:47

79) Friedman number

A Friedman number is an integer, which represented in a given numeral system, is the result of a non-trivial expression using all its own digits in combination with any of the four basic arithmetic operators

), additive inverses, parentheses, exponentiation, and concatenation. Here, non-trivial means that at least one operation besides concatenation is used. Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 024 = 20 + 4. For example, 347 is a Friedman number in the decimal numeral system, since

The decimal 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, ....

Friedman numbers are named after Erich Friedman, a now-retired mathematics professor at Stetson University and recreational mathematics enthusiast.

A Friedman prime is a Friedman number that is also prime. The decimal Friedman primes are:

127, 347, 2503, 12101, 12107, 12109, 15629, 15641, 15661, 15667, 15679, 16381, 16447, 16759, 16879, 19739, 21943, 27653, 28547, 28559, 29527, 29531, 32771, 32783, 35933, 36457, 39313, 39343, 43691, 45361, 46619, 46633, 46643, 46649, 46663, 46691, 48751, 48757, 49277, 58921, 59051, 59053, 59263, 59273, 64513, 74353, 74897, 78163, 83357, ...

The expressions of the first few Friedman numbers are

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

The first nice Friedman numbers are:

127, 343, 736, 1285, 2187, 2502, 2592, 2737, 3125, 3685, 3864, 3972, 4096, 6455, 11264, 11664, 12850, 13825, 14641, 15552, 15585, 15612, 15613, 15617, 15618, 15621, 15622, 15623, 15624, 15626, 15632, 15633, 15642, 15645, 15655, 15656, 15662, 15667, 15688, 16377, 16384, 16447, 16875, 17536, 18432, 19453, 19683, 19739 ...

A nice Friedman prime is a nice Friedman number that's also prime. The first nice Friedman primes are:

127, 15667, 16447, 19739, 28559, 32771, 39343, 46633, 46663, 117619, 117643, 117763, 125003, 131071, 137791, 147419, 156253, 156257, 156259, 229373, 248839, 262139, 262147, 279967, 294829, 295247, 326617, 466553, 466561, 466567, 585643, 592763, 649529, 728993, 759359, 786433, 937577.

Michael Brand proved that the density of Friedman numbers among the naturals is 1, which is to say that the probability of a number chosen randomly and uniformly between 1 and n to be a Friedman number tends to 1 as n tends to infinity. This result extends to Friedman numbers under any base of representation. He also proved that the same is true also for binary, ternary and quaternary nice Friedman numbers. The case of base-10 nice Friedman numbers is still open.

Vampire numbers are a subset of Friedman numbers where the only operation is a multiplication of two numbers with the same number of digits, for example

.

Jai Ganesh · 2024-07-21 17:59:49

80) Powers

Jai Ganesh · 2024-07-21 18:26:48

81) Tenth Power of the First Ten Natural Numbers

,
,
,
,
,
,
,
,
,
.

Jai Ganesh · 2024-07-22 23:11:12

82)

.

Jai Ganesh · 2024-07-24 23:04:42

83)

.

Jai Ganesh · 2024-08-01 21:55:03

84)

Jai Ganesh · 2024-08-09 20:56:59

85) Regular Number

Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors are 2, 3, and 5. As an example,

,
so as divisors of a power of 60 both 48 and 75 are regular.

These numbers arise in several areas of mathematics and its applications, and have different names coming from their different areas of study.

In number theory, these numbers are called 5-smooth, because they can be characterized as having only 2, 3, or 5 as their prime factors. This is a specific case of the more general k-smooth numbers, the numbers that have no prime factor greater than k.

In the study of Babylonian mathematics, the divisors of powers of 60 are called regular numbers or regular sexagesimal numbers, and are of great importance in this area because of the sexagesimal (base 60) number system that the Babylonians used for writing their numbers, and that was central to Babylonian mathematics.
In music theory, regular numbers occur in the ratios of tones in five-limit just intonation. In connection with music theory and related theories of architecture, these numbers have been called the harmonic whole numbers.
In computer science, regular numbers are often called Hamming numbers, after Richard Hamming, who proposed the problem of finding computer algorithms for generating these numbers in ascending order. This problem has been used as a test case for functional programming.

Number theory

Formally, a regular number is an integer of the form

, for nonnegative integers i, j, and k. Such a number is a divisor of
. The regular numbers are also called 5-smooth, indicating that their greatest prime factor is at most 5. More generally, a k-smooth number is a number whose greatest prime factor is at most k.

The first few regular numbers are

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, ...

Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers. A regular number

is less than or equal to some threshold N if and only if the point
belongs to the tetrahedron bounded by the coordinate planes and the plane
as can be seen by taking logarithms of both sides of the inequality
. Therefore, the number of regular numbers that are at most
N can be estimated as the volume of this tetrahedron, which is
.

Jai Ganesh · 2024-08-27 17:43:02

86) Concrete Number

A concrete number or numerus numeratus is a number associated with the things being counted, in contrast to an abstract number or numerus numerans which is a number as a single entity. For example, "five apples" and "half of a pie" are concrete numbers, while "five" and "one half" are abstract numbers. In mathematics the term "number" is usually taken to mean an abstract number. A denominate number is a type of concrete number with a unit of measure attached with it. For example, "5 inches" is a denominate number because it has the unit inches after it.

History

Mathematicians in ancient Greece were primarily interested in abstract numbers, while writers of instructional books for practical use were not concerned with such distinctions, so the terminology distinguishing the two types of number was slow to appear. In the 16th century textbooks began to make the distinction. This has appeared with increasing frequency until modern times.

Denominate numbers

Denominate numbers are further classified as either simple, meaning a single unit is given, or compound, meaning multiple units are given. For example, 6 kg is a simple denominate number, while 324 yards 1 foot 8 inches is a compound denominate number. The process of converting a denominate number to an equivalent form that uses a different unit is called reduction. More specifically, reduction to a lower or higher unit of measurement is called reduction to lower or higher denominations. Reduction to a lower denomination is accomplished by multiplying by the number of lower units contained in each higher unit. In the case of a compound denominant number, the products are then added together. For example, 1 hour 23 minutes 20 seconds is

Similarly, a division is used to reduce to a higher denomination, and remainders can be applied to the next highest unit to form compound denominant numbers. Addition and subtraction of compound numbers can be performed by grouping the amounts associated with each unit and performing the necessary carry and borrow operations. Multiplication and division by a pure number are again similar.

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