1,2

The aliquot sequence for n is the trajectory of n under repeated application of the map x -> sigma(x) - x.

The trajectory will either have a transient part followed by a cyclic part, or will have an infinite transient part and never cycle.

Sequence gives (length of transient part of trajectory) + (length of cycle if the trajectory did not reach 0). In other words here we consider that the trajectory ends if we reach 1.

Given that sigma(n) is the sum of the divisors of n which are less than n, we have that the aliquot length A(n) = r-1 where r is the smallest integer such that sigma^r(n) = sigma^s(n) for some s<r. If this never happens (i.e. if r is infinite) then the sequence has length 0.

In the interval [1,1000] it is not known if the aliquot length is 0 for the numbers 276, 552, 564, 660 and 966.

T. D. Noe, Table of n, a(n) for n=1..275

Author?, Prime Families

Eric Weisstein's World of Mathematics, Aliquot Sequence

P. Zimmermann, Aliquot Sequences

a(12) = 7: 12 is divisible by 1,2,3,4 and 6 so sigma(12)=16; 16 is divisible by 1,2,4 and 8 so sigma(16)=15; 15 is divisible by 1,3 and 5 so sigma(15)=9; 9 is divisible by 1 and 3 so sigma(9)=4; 4 is divisible by 1 and 2 so sigma(4)=3; 3 is divisible only by 1 so sigma(3)=1; 1 is not divisible by anything less than one so sigma(1)=0. The aliquot sequence is therefore 16, 15, 9, 4, 3, 1, 0 which is 7 elements long. Therefore a(12) = 7.

See A098007, A003023 for other versions. See A008886 for the aliquot sequence of 42.

Sequence in context: A076494 A071862 A030362 this_sequence A096826 A116199 A162915

Adjacent sequences: A044047 A044048 A044049 this_sequence A044051 A044052 A044053

nonn,nice

Aza Raskin (aza(AT)uchicago.edu), Jun 25 2003