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.

]]>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

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

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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

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

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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

.]]>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 orAs

is also equal to 4, or can thus be written as , oror as the pentation,

(hyperoperation notation).

65536 is a superperfect number - a number such that

.]]>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.