Sequence A049096 with Counterexample or New Definition
Rainer Rosenthal
r.rosenthal at web.de
Tue Nov 1 11:22:16 CET 2005
The number k=182 is missing in
http://www.research.att.com/projects/OEIS?Anum=A049096
because 2^k+1 = 0 (mod 1093^2) and the definition is
"**2^n + 1 is divisible by a square"
Possibility #1: stick to the definition, enhancing
it slightly to "2^a(n) + 1 is divisible by a square"
and inserting the number 182 between 177 and 183.
Possibility #2: change the definition according
to the author's (Vladeta Jovovic) conjecture:
"2^n+1 is square-free iff gcd(n,2^n+1)=1"
which is proven wrong by n=182.
This will give the longer but correct definition:
"2^a(n) + 1 is divisible by a square and coprime to a(n)"
My favorite is possibility #1 with a comment regarding
the exceptional value a(43) = 182.
....,,;;---''^^^''---;;,,,....
And there should be a cross link sequences A086982, which
is just the same with base b=10 instead of b=2:
A086982: "a(n) = k where (10^k+1) is not square-free."
This sequence *does* contain the exceptional value, which
is 243: 10^243+1 = 0 (mod 487^2). Isn't it funny:
A086982 contains an exceptional
value (243) but without comment.
It is linked to A086981, which
*has* a comment for a(93)=243
but doesn't contain this value
(showing only 54 members).
I don't feel competent enough for the appropriate changes
in this area. So may be one of the editors of the OEIS
could be so kind to do so?
Best regards,
Rainer Rosenthal
r.rosenthal at web.de
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