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2018-04-02, 13:23   #11
10metreh

Nov 2008

44228 Posts

Quote:
 Originally Posted by shortcipher The numbers are taken from an aliquot sequence project I am running and don't necessarily relate to any existing factors in the database.
The thing is, they do.

It appears that you are computing the aliquot sequence of 2^(p-1)*(2^p-1)*3 for some Mersenne prime 2^p-1. If a has no common factors with 2^(p-1)*(2^p-1), then σ(2^(p-1)*(2^p-1)*a) = σ(2^(p-1)*(2^p-1))*σ(a) = 2^p*(2^p-1)*σ(a). Hence if 2^(p-1)*(2^p-1)*a is a term in your aliquot sequence, then the next term is 2^(p-1)*(2^p-1)*(2σ(a)-a).
But the value 2σ(a)-a does not depend on p, so for each Mersenne prime, 2^(p-1)*(2^p-1)*3 has essentially the same aliquot sequence, with just the power of two and the Mersenne prime differing. The pattern only breaks when a term happens to have a second factor of 2^p-1, i.e. the value a above has a common factor with 2^(p-1)*(2^p-1). This is very unlikely for large p.

For 2^13-1 and all greater Mersenne primes, we have not yet found such a term, so all of these sequences are still on the same trajectory. Thus when we compute a new term of the aliquot sequence of 2^12*(2^13-1)*3, we get a new term of the sequence of 2^(p-1)*(2^p-1)*3 for all larger Mersenne primes.

The aliquot sequence of 2^12*(2^13-1)*3 (http://factordb.com/sequences.php?se...t20&fr=1&to=20) is known up to index 863; the factors you posted earlier in the thread come from terms 817 and 818, so whichever Mersenne prime you were actually using, you were in fact redoing work that has already been done.