Template:Sequence of the Day for July 27

Intended for: July 27, 2012

Timetable

• First draft entered by Alonso del Arte on May 5, 2011 ✓
• Draft reviewed by Alonso del Arte on July 22, 2011 ✓
• Draft approved by Daniel Forgues on July 24, 2011 ✓

Yesterday's SOTD * Tomorrow's SOTD

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A008892: Aliquot sequence starting at 276.

{ 276, 396, 696, 1104, 1872, 3770, 3790, ... }

When The Encyclopedia of Integer Sequences was published in 1995, it was believed that there exist numbers such that their aliquot sequences neither terminate nor eventually cycle (the aliquot sequences for perfect numbers, amicable pairs and sociable numbers of order t [with t = 1, t = 2 and t > 2 respectively] very obviously cycle, while others may take a dozen or so terms to reach 1 [after hitting a prime number] which begets the empty sum, i.e. 0, and the sequence terminates) but no example was known. D. H. Lehmer succeeded in ruling out the first thousand positive integers except for 276, 552, 564, 660 and 966. Today, 276 still looks like a good candidate for a number with an aliquot sequence that neither terminates nor eventually cycles!? According to FactorDB.com, the last term in this sequence to be factored and checked is a (1663), a number which has among its proper divisors 2, 3 2, 7, 9551, 26441307811, and 72269874661669001519651. Any non-terminating aliquot sequence which never cycle would have to manage to never hit neither a previous term nor any prime number while growing (monotonically or not) without bounds!