Aliquot sequence

In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum for 1 is 0. The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ₁ or the aliquot sum function s in the following way:^[1]

s₀ = k

s_n = s(s_n−1) = σ₁(s_n−1) − s_n−1 if s_n−1 > 0,

and s(0) is undefined.

For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because:

σ₁(10) − 10 = 5 + 2 + 1 = 8,

σ₁(8) − 8 = 4 + 2 + 1 = 7,

σ₁(7) − 7 = 1,

σ₁(1) − 1 = 0.

Many aliquot sequences terminate at zero; all such sequences necessarily end with a prime number followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no proper divisors). See (sequence A080907 in the OEIS) for a list of such numbers up to 75. There are a variety of ways in which an aliquot sequence might not terminate:

• A perfect number has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6, 6, 6, 6, ...
• An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is 220, 284, 220, 284, ...
• A sociable number has a repeating aliquot sequence of period 3 or greater. (Sometimes the term sociable number is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is 1264460, 1547860, 1727636, 1305184, 1264460, ...
• Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable. For instance, the aliquot sequence of 95 is 95, 25, 6, 6, 6, 6, ... . Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers ( A063769).

The lengths of the Aliquot sequences that start at n are

1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 1, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, ... (sequence A044050 in the OEIS)

The final terms (excluding 1) of the Aliquot sequences that start at n are

1, 2, 3, 3, 5, 6, 7, 7, 3, 7, 11, 3, 13, 7, 3, 3, 17, 11, 19, 7, 11, 7, 23, 17, 6, 3, 13, 28, 29, 3, 31, 31, 3, 7, 13, 17, 37, 7, 17, 43, 41, 3, 43, 43, 3, 3, 47, 41, 7, 43, ... (sequence A115350 in the OEIS)

Numbers whose Aliquot sequence terminates in 1 are

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... (sequence A080907 in the OEIS)

Numbers whose Aliquot sequence terminates in a perfect number are

25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913, ... (sequence A063769 in the OEIS)

Numbers whose Aliquot sequence terminates in a cycle with length at least 2 are

220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, ... (sequence A121507 in the OEIS)

Numbers whose Aliquot sequence is not known to be finite or eventually periodic are

276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, ... (sequence A131884 in the OEIS)

An important conjecture due to Catalan is that every aliquot sequence ends in one of the above ways: with a prime number, a perfect number, or a set of amicable or sociable numbers.^[2] The alternative would be that a number exists whose aliquot sequence is infinite yet never repeats. Any one of the many numbers whose aliquot sequences have not been fully determined might be such a number. The first five candidate numbers are often called the Lehmer five (named after D.H. Lehmer): 276, 552, 564, 660, and 966.^[3]

As of April 2015, there were 898 positive integers less than 100,000 whose aliquot sequences have not been fully determined, and 9190 such integers less than 1,000,000.^[4]

Systematically searching for aliquot sequences

The aliquot sequence can be represented as a directed graph, $G_{n,s}$, for a given integer $n$, where $s(k)$ denotes the sum of the proper divisors of $k$.^[5] Cycles in $G_{n,s}$ represent sociable numbers within the interval $[1,n]$. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.

Notes

1. Weisstein, Eric W. "Aliquot Sequence". MathWorld.
2. Weisstein, Eric W. "Catalan's Aliquot Sequence Conjecture". MathWorld.
3. Creyaufmüller, Wolfgang (May 24, 2014). "Lehmer Five". Retrieved June 14, 2015.
4. Creyaufmüller, Wolfgang (April 29, 2015). "Aliquot Pages". Retrieved June 14, 2015.
5. Rocha, Rodrigo Caetano; Thatte, Bhalchandra (2015), Distributed cycle detection in large-scale sparse graphs (PDF), Simpósio Brasileiro de Pesquisa Operacional (SBPO)

References

• Manuel Benito; Wolfgang Creyaufmüller; Juan Luis Varona; Paul Zimmermann. Aliquot Sequence 3630 Ends After Reaching 100 Digits. Experimental Mathematics, vol. 11, num. 2, Natick, MA, 2002, p. 201-206.
• W. Creyaufmüller. Primzahlfamilien - Das Catalan'sche Problem und die Familien der Primzahlen im Bereich 1 bis 3000 im Detail. Stuttgart 2000 (3rd ed.), 327p.

External links

• Tables of Aliquot Cycles (J.O.M. Pedersen)
• Aliquot Page (Wolfgang Creyaufmüller)
• Aliquot sequences (Christophe Clavier)
• Forum on calculating aliquot sequences (MersenneForum)
• Aliquot sequence summary page for sequences up to 100000 (there are similar pages for higher ranges) (Karsten Bonath)
• Active research site on aliquot sequences (Jean-Luc Garambois)