What is Primitive abundant number?
In [[Mathematics|math]], a '''primitive abundant number''' is a special kind of [[abundant number]]. Its proper [[Division (mathematics)|divisors]], however, must all be deficient numbers(numbers whose sum of proper [[Division (mathematics)|divisors]] are less than 2 times that number).<ref>{{MathWorld|title=Primitive Abundant Number}}</ref><ref>Erdős adopts a wider definition that requires a primitive abundant number to be not deficient, but not necessarily abundant (Erdős, Surányi and Guiduli. ''Topics in the Theory of Numbers'' p214. Springer 2003.). The Erdős definition allows [[Perfect number|perfect numbers]] to be primitive abundant numbers too.</ref> ==Example== For example, 20 is a primitive abundant number because: # 20 is an [[abundant number]]. This is because the [[sum]] of its [[Divisor|divisors]] is 1 + 2 + 4 + 5 + 10 + 20 > 40. This makes 20 an [[abundant number]]. # The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8. All of these numbers are a deficient number. This makes 20 a primitive abundant number. The first few primitive abundant numbers are [[20 (number)|20]], [[70 (number)|70]], [[88 (number)|88]], [[104 (number)|104]], 272, 304, 368, 464, 550, 572 ... {{OEIS|id=A071395}} The smallest [[Odd number|odd]] primitive abundant number is 945. == Properties == Every [[multiple]] of a primitive abundant number is [[Abundant number|abundant]] . Every [[abundant number]] is a [[multiple]] of either a primitive abundant number or a [[multiple]] of a [[perfect number]]. Every primitive abundant number is either a primitive semiperfect number or a weird number. There is an [[Infinity|infinite]] amount of primitive abundant numbers.<ref>Paul Erdős, ''Journal of the London Mathematical Society'' 9 (1934) 278&#x2013;282.</ref> == References == {{reflist}} {{Classes of natural numbers}} {{Divisor classes}} [[Category:Divisor function]] [[Category:Integer sequences]]