Large Clubs
It is an open problem whether there are infinitely large clubs of mutually friendly numbers. The perfect numbers form a club, and it is conjectured that there are infinitely many perfect numbers (at least as many as there are Mersenne primes), but no proof is known. As of June 2009, 47 perfect numbers are known, the largest of which has more than 25 million digits in decimal notation. There are clubs with more known members, in particular those formed by multiply perfect numbers, which are numbers whose abundancy is an integer. As of early 2011, the club of friendly numbers with abundancy equal to 9 has 2091 known members. Although some are known to be quite large, clubs of multiply perfect numbers (excluding the perfect numbers themselves) are conjectured to be finite.
Read more about this topic: Friendly Number
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