Semiperfect number

Semiperfect number

In mathematics, a semiperfect number or pseudoperfect number is a natural number "n" that is equal to the sum of all or some of its proper divisors.

The first few semiperfect numbers are:6, 12, 18, 20, 24, 28, 30, 36, 40, ... OEIS|id=A005835;every multiple of a semiperfect number is semiperfect, and every number of the form 2"m""p" for a natural number "m" and a prime number "p" such that "p" < 2"m" + 1 is also semiperfect.

The smallest odd semiperfect number is 945 (see, e.g., Friedman 1993).

A semiperfect number that is equal to the sum of all its proper divisors is called a perfect number; an abundant number which is not semiperfect is called a weird number. With the exception of 2, all primary pseudoperfect numbers are semiperfect. Every practical number that is not a power of two is semiperfect.

A semiperfect number that is not divisible by any smaller semiperfect number is a primitive semiperfect number.

References

* cite journal
title = Sums of divisors and Egyptian fractions
author = Friedman, Charles N.
journal = Journal of Number Theory
year = 1993
volume = 44
pages = 328–339
url = http://dell5.ma.utexas.edu/users/friedman/divisors.ps
id = MathSciNet | id = 1233293
doi = 10.1006/jnth.1993.1057

External links

* [http://mathworld.wolfram.com/SemiperfectNumber.html MathWorld: Semiperfect number]


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Primitive semiperfect number — In mathematics, a primitive semiperfect number (also called a primitive pseudoperfect number , irreducible semiperfect number or irreducible pseudoperfect number ) is a semiperfect natural number that has no semiperfect proper divisor.The first… …   Wikipedia

  • List of recreational number theory topics — This is a list of recreational number theory topics (see number theory, recreational mathematics). Listing here is not pejorative: many famous topics in number theory have origins in challenging problems posed purely for their own sake. See list… …   Wikipedia

  • Prime number — Prime redirects here. For other uses, see Prime (disambiguation). A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is… …   Wikipedia

  • 300 (number) — This article is about the numbers 300 to 399. For other uses of 300, see 300 (disambiguation). For the guitar, see Gibson ES 335. For the British tilting train, see British Rail Class 390. For the Dada magazine, see 391 (magazine). For the… …   Wikipedia

  • Perfect number — In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect… …   Wikipedia

  • Composite number — A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number. So, if n > 0 is an integer and… …   Wikipedia

  • Highly composite number — This article is about numbers having many divisors. For numbers factorized only to powers of 2, 3, 5 and 7 (also named 7 smooth numbers), see Smooth number. A highly composite number (HCN) is a positive integer with more divisors than any… …   Wikipedia

  • Multiply perfect number — In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. For a given natural number k, a number n is called k perfect (or k fold perfect) if and only if the sum of… …   Wikipedia

  • Deficient number — In number theory, a deficient number or defective number is a number n for which the sum of divisors σ(n)<2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)<n. The value 2n − σ(n) (or n − s(n)) is… …   Wikipedia

  • Harmonic divisor number — This article is about harmonic divisor numbers. For meanings of harmonic number, see harmonic number (disambiguation). In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”