[seqfan] Re: Pairs Occurring Only Once Among # Of Divisors
Maximilian Hasler
maximilian.hasler at gmail.com
Fri Jun 26 05:23:24 CEST 2009
Proof of 65535 and 65536 :
d(65535)=16
d(65536)=17
d(n+1)=17 <=> n+1 = p^16, p prime.
But then
n = p^16-1 = (p-1)(p+1)(p^2+1)(p^4+1)(p^8+1)
and for p>2,
n = 2a 2b 2c 2d 2e with a,b,c,d,e odd
thus numdiv(n) >= numdiv(2^5*3^5) = 36
(a=1 <=> p=3 also gives more divisors, as I could have mentioned
explicitely in the earlier posts.)
So p=2 is the only possibility, and 65535 is in the sequence.
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d(65536)=17
d(65537)=2
oh, I see this is already in Richard's draft...
Maximilian
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