[seqfan] Re: Observation on the divisors of 136

Brad Klee bradklee at gmail.com
Mon Jan 15 06:33:01 CET 2018


After: http://list.seqfan.eu/pipermail/seqfan/2017-December/018225.html
And: http://list.seqfan.eu/pipermail/seqfan/2018-January/018249.html

I'm not sure what this latest theorem means, and confused by term
"prefixes". However, the details seem to reiterate and / or improve
assertions from messages last year. Ruling out additional prime factors is
essentially the same as ruling out additional ones in the binary.  Again
assume two prime factors p1, p2.

n  = p1*p2*(2^k);
2^k < p1 < p2 < p1*p2.

This requires,

p1 = 2^(k+1) + 1
p2 = p1*2^(k+1) + 1 = p1*(p1-1) + 1
p1*p2 = p2*2^(k+1) + 1 = p2*(p1-1) + 1

The constraint on p1*p2 ( I neglected to calculate this earlier ) leads
readily to impossibility, p2 = 1. No two prime factors, no three ones in
binary. Then the proposed sequence possibly reduces to a union of
A191363(?) and A000079. Antti Karttunen's suggestion to look at A295296
provokes more curiosity. The appearance of odd 315 remains part of an
unexplained mystery.

Cheers,

Brad


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