ddrummer realtime

free counters

Wednesday, December 28, 2011

Interesting facts about the positive integers from 1 to 10000



Compute Divisors for Positive Integers

Use the form below to get the divisors of and additional information about any positive integer up to 1 million. (decimal places are rounded)

Or just append the integer you want information about to the end of this site's URL. e.g., www.positiveintegers.org/5

   

The Integers 1 to 10000

  • Range is a range of numbers, in groups of 100. Click on the range for more information about that range.
  • Count(Primes) is the count of Prime Numbers in that range.
  • Count(Fibonacci) is the count of Fibonacci Numbers in that range.
  • Max(Count(d(N))) is the highest number of divisors that any single number within that range possesses.
  • Most Composite N is the list of the numbers in the range that have the most divisors.
  • Count(Deficient)Count(Abundant), and Count(Perfect) are the counts ofDeficientAbundant, and Perfect numbers in that range.
RangeCount(Primes)Count(Fibonacci)Max(Count(d(N)))Most Composite NCount(Deficient)Count(Abundant)Count(Perfect)
1-10025101260, 72, 84, 90, 9676222
101-2002111818076240
201-3001612024077230
301-4001612436073270
401-50017024420, 48074251
501-60014024504, 540, 60076240
601-70016124630, 660, 67276240
701-8001403072074260
801-9001503284075250
901-10001412896074260
1001-110016032108077230
1101-120012030120076240
1201-130015036126076240
1301-140011032132074260
1401-150017036144074260
1501-1600121321512, 156077230
1601-170015040168074260
1701-180012036180075250
1801-1900120321848, 189076240
1901-200013036198074260
2001-2100140362016, 210074260
2101-220010040216076240
2201-230015032228075250
2301-2400150362340, 240077230
2401-250010030244874260
2501-260011148252074260
2601-270015040264078220
2701-280014036277274260
2801-290012042288075250
2901-300011036294074260
3001-310012040302476240
3101-320010040312076240
3201-330011040324074260
3301-340015048336074260
3401-350011036342074260
3501-360014045360077230
3601-370013040369678220
3701-380012048378073270
3801-3900110363840, 390074260
3901-400011048396075250
4001-410015042403274260
4101-42009148420076240
4201-430016036428474260
4301-44009048432077230
4401-4500110364410, 450077230
4501-4600120404536, 456073270
4601-4700120484620, 468074260
4701-480012042480074260
4801-490080364860, 489676240
4901-500015036495077230
5001-510012060504075250
5101-520011036514874260
5201-530010048528077230
5301-540010048540074260
5401-550013048546075250
5501-560013048554474260
5601-5700120405616, 567075250
5701-580010048576076240
5801-590016048588073270
5901-60007048594074260
6001-610012048604876240
6101-620011048612077230
6201-630013054630072280
6301-640015042633675250
6401-65008050648074260
6501-6600110486552, 660076240
6601-6700100366624, 666073270
6701-680012156672076240
6801-690012048684076240
6901-700013048693074260
7001-71009048702076240
7101-720010054720074260
7201-730011040728074260
7301-74009048739276240
7401-750011042748875250
7501-760015064756073270
7601-770012040768076240
7701-780010048780077230
7801-7900100367812, 784075250
7901-800010060792074260
8001-810011048806472280
8101-8200100488160, 819078211
8201-830014048828074260
8301-84009060840077230
8401-85008040842473270
8501-8600120488568, 858074260
8601-870013056864078220
8701-880011048873675250
8801-890013054882075250
8901-90009048900076240
9001-910011050907276240
9101-9200120489120, 918073270
9201-930011064924074260
9301-940011060936075250
9401-950015048945075250
9501-960070489504, 9576, 960075250
9601-970013048966076240
9701-980011048972073270
9801-990012054990078220
9901-100009040993677230

The Integers

The integers consist of the positive natural numbers (1, 2, 3, …) thenegative natural numbers (-1, -2, -3, ...) and the number zero. The set of all integers is usually denoted in mathematics by Z - The Integers in blackboard bold, which stands for Zahlen (German for "numbers").

Positive Integers refers to all whole number greater than zero. Zero is not a positive integer. For each positive integer there is a negative integer. Integers greater than zero are said to have a positive "sign".

The Positive Integers are a subset of the Natural Numbers (N - The Natural Numbers), depending on whether or not 0 is considered a Natural Number. The term Positive Integers is preferred over Natural Numbers and Counting Numbers because it is more clearly defined; there is inconsistency over whether zero is a member of those sets. Zero is not an element of the Positive Integers.

The Positive Integers are symbolized by Z+ - The Positive Integers.

Prime numbers are a subset of the positive integers and are of special interest in Number Theory. Note that the number 1 is not a prime number; i.e., for the set of prime numbers P - The Prime Numbers, all P - The Prime Numbers > 1. A prime number is a positive integer that has no positive integer divisors except for 1 and itself. Positive Integers that are not Prime Numbers or 1 are Composite Numbers. The number 1 is neither a Prime Number nor a Composite Number.

Algebraic properties of Integers

Like the natural numbers, Z - The Integers is closed under the operations of addition andmultiplication; that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers and zero, Z - The Integers(unlike the natural numbers) is also closed under subtractionZ - The Integers is not closed under the operation of division, since the quotient of two integers ( e.g. , 1 divided by 2), need not be an integer.

The following table lists some of the basic properties of addition and multiplication for any integers a , b and c .

additionmultiplication
closure :a  +  b    is an integera  ×  b    is an integer
associativity :a  + ( b  +  c )  =  ( a  +  b ) +  ca  × ( b  ×  c )  =  ( a  ×  b ) ×  c
commutativity :a  +  b   =   b  +  aa  ×  b   =   b  ×  a
existence of an identity element :a  + 0  =   aa  × 1  =   a
existence of inverse elements :a  + (- a )  =  0
distributivity :a  × ( b  +  c )  =  ( a  ×  b ) + ( a  ×  c )

Ordering

Z - The Integers is a totally ordered set without an upper or lower bound. The ordering of Z - The Integersis given by

... < -2 < -1 < 0 < 1 < 2 < ...

An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive.

The ordering of integers is compatible with the algebraic operations in the following way:

  1. if a < b and c < d , then a + c < b + d
  2. if a < b and 0 < c , then ac < bc

Divisors

divisor of an integer n, also called a factor of n, is an integer which evenly divides without leaving a remainder. If x is a divisor of n, it can be written that x|n. This is read as x divides n. It is also said that n is divisible by x, and that n is a multiple of x.

1 and -1 are divisors of every integer, and every integer is a divisor of 0. Numbers divisible by 2 are called even and those that are not are called odd.

Blog Archive