1,2

Various starting values, which are even or reach an even value, follow the associated trajectory of the standard aliquot sequence A098007 and thus grow to values whose ultimate fate is currently unknown. The first such term is 165 as its cycle begins 165 -> 288 -> 531 -> 780, from which point it follows the aliquot sequence starting at 780 that after 3485 further cycles reaches a term with 198 digits whose factorization is not known.

This sequence also has the same loops seen in the aliquot sequence. All even perfect numbers repeat after one value, and all even-numbered amicable pairs and even-numbered sociable number loops are also present. The amicable pair 2620-2934 is first reached from n = 477 after 329 cycles while the pair 1184-1210 is first reached from n = 847 after 4 cycles. The 28-member sociable number loop containing the number 376736, see A072890, is first reached from n = 267 after 52 cycles.

Loops present in this sequence that are not in the aliquot sequence are two-member loops formed by a Mersenne prime and the even number, a power of 2, one greater than it. For starting values n <= 1000 two other loops are present which end numerous cycles. One is a five-member loop comprising the numbers 56 -> 64 -> 63 -> 104 -> 106 (-> 56), first seen for starting value n = 11. The other is a four-member loop comprising the numbers 40 -> 50 -> 43 -> 44 (-> 40), first seen for starting value n = 27. It is likely other such loops exist for starting values >> 1000.

As sigma(prime) = prime + 1, such prime starting values will have the same cycle, with one extra term, as the later even term, assuming the prime is not a Mersenne prime.

For the first 1000 terms the longest currently known cycle if one of 1538 steps, starting at n = 693 and ending with 104. The largest value reached in this series is a 58-digit number 12316...65104. The majority of this series follows the 1074-step aliquot sequence starting at 1248.

Wikipedia, Aliquot sequence

P. Zimmermann, Aliquot Sequences

a(3) = 2 as 3 -> 4 -> 3. Similarly for all other Mersenne primes.

a(5) = 2 as 5 -> 6 -> 6. Similarly for all other primes one less than a perfect number.

a(6) = 1 as 6 -> 6. Similarly for all other even perfect numbers.

a(11) = 245 as 11 -> 12 -> 16 -> 15 -> 24 -> (233 more terms) -> 230 -> 202 -> 104 -> 106 -> 56 -> 64 -> 63 -> 104, ending with the five-member loop.

a(19) = 7 as 19 -> 20 -> 22 -> 14 -> 10 -> 8 -> 7 -> 8.

a(27) = 5 as 27 -> 40 -> 50 -> 43 -> 44 -> 40, ending with the four-member loop.

a(847) = 5 as 847 -> 1064 -> 1336 -> 1184 -> 1210 -> 1184, ending with the second smallest amicable pair.

Table[t=k=0; lst={n}; k=If[OddQ@n, DivisorSigma[1, n], DivisorSigma[1, n]-n]; While[FreeQ[lst, k], AppendTo[lst, k]; n=k; t++; k=If[OddQ@n, DivisorSigma[1, n], DivisorSigma[1, n]-n]]; t+1, {n, 100}] (* Giorgos Kalogeropoulos, Aug 14 2021 *)

Cf. A098007, A001065, A259180, A003416, A072890, A000668.

Sequence in context: A052435 A094701 A210452 * A240301 A289641 A209312

Adjacent sequences: A347034 A347035 A347036 * A347038 A347039 A347040

Scott R. Shannon, Aug 12 2021

approved