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In mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers.
For example, 20 is a primitive abundant number because:
- The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number.
- The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number.
The first few primitive abundant numbers are:
20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ... (sequence A071395 in the OEIS)The smallest odd primitive abundant number is 945.
A variant definition is abundant numbers having no abundant proper divisor (sequence A091191 in the OEIS). It starts:
12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114Properties
Every multiple of a primitive abundant number is an abundant number.
Every abundant number is a multiple of 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 are an infinite number of primitive abundant numbers.
The number of primitive abundant numbers less than or equal to n is