Multiply perfect number

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 all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of July 2004, k-perfect numbers are known for each value of k up to 11.

It can be proven that:

  • For a given prime number p, if n is p-perfect and p does not divide n, then pn is (p+1)-perfect. This implies that an integer n is a 3-perfect number divisible by 2 but not by 4, if and only if n/2 is an odd perfect number, of which none are known.
  • If 3n is 4k-perfect and 3 does not divide n, then n is 3k-perfect.

Smallest k-perfect numbers

The following table gives an overview of the smallest k-perfect numbers for k <= 7 (cf. Sloane's A007539):

k Smallest k-perfect number Found by
1 1 ancient
2 6 ancient
3 120 ancient
4 30240 René Descartes, circa 1638
5 14182439040 René Descartes, circa 1638
6 154345556085770649600 Robert Daniel Carmichael, 1907
7 141310897947438348259849402738 485523264343544818565120000 TE Mason, 1911

For example, 120 is 3-perfect because the sum of the divisors of 120 is
1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = 360 = 3 × 120.

External links


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • 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

  • number game — Introduction       any of various puzzles and games that involve aspects of mathematics.       Mathematical recreations comprise puzzles and games that vary from naive amusements to sophisticated problems, some of which have never been solved.… …   Universalium

  • Friendly number — In number theory, a friendly number is a natural number that shares a certain characteristic called abundancy, the ratio between the sum of divisors of the number and the number itself, with one or more other numbers. Two numbers with the same… …   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

  • 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

  • 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

  • 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

  • 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

  • Pronic number — A pronic number, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive integers, that is, n (n + 1). The n th pronic number is twice the n th triangular number and n more than… …   Wikipedia

Share the article and excerpts

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