A unitary aliquot sequence is constructed similarly to a "standard" aliquot sequence, except that instead of adding divisors, one adds up unitary divisors. A unitary divisor D of a number N satisfies two requirements: D divides evenly into N, and D and N have no common divisor other than 1.
It is conjectured that all unitary aliquot sequences eventually either enter a closed cycle, or terminate by reaching the number 1. Certainly no counterexample is known, and the author has now verified the conjecture for all starting numbers up to 400000000.
This new "world record" unitary aliquot sequence begins with the number 151244562, and terminates 16657 steps later by reaching the number 1.
The largest member of the sequence occurs at step 4641, and is a 90-digit number:
C90 = 987891251357667899392132479835603514727233722302097859935661683164969735652342467298231062
C90 = 2 * 3 * 3 * 3 * 3 * 503 * 9682217399 * P76
P76 = 1252135446163593391182747087802437681221287268786627017667109514519500763883
The most "difficult" factorization encountered during the computation of this sequence is the factorization of the number at step 4723:
C89 = 26517999942774974531899973202351905464540371267317651256256008854551681034056102250308158
C89 = 2 * 3 * P43 * P45
P43 = 6693563560006615522852930516678376286981823
P45 = 660286052042029804004217680192144404744215691
In all, there were 40488 distinct prime factors needed to complete this sequence, and thus 40488 numbers were proven prime. The largest prime factor occurs in the factorization of the number at step 4680:
C90 = 321676783420213779760115268514028350660244936661818176222251840466802608942243730745057786
C90 = 2 * 3 * P89
P89 = 53612797236702296626685878085671391776707489443636362703708640077800434823707288457509631
The computation of the sequence occurred over a period of about 29 weeks, using one 450 MHz Pentium-II on a full-time basis, and three other similar machines on a part-time basis. The primary software tools were a modified implementation of GMPFAC (written by Conrad Curry), and the PPMPQS program (written by Satoshi Tomabechi). The GMPFAC implementation was modified to use the PPMPQS program for splitting difficult composites if the ECM method did not yield a factor within a certain timeframe.
All prime factors were proven probable prime using GMPFAC; upon completion of the sequence, PPMPQS was used to rigorously prove primality for all but two of the prime factors. PPMPQS was unable to prove primality for two of the factors: 2 and 19004056835230495043. The proof that 2 is prime should be obvious to any reader :-) The other number was proven prime using an N-1 factorization; PPMPQS was unable to prove it prime - the author of PPMPQS has since found the problem and will soon release an updated version of the program.
The complete sequence along with prime factorizations
of each element (gzipped) - over 900K on a slow link - 1 minute download time
Please email comments to Jack Brennen