[seqfan] Re: More composite numbers are needed for A226181

[Max Alekseyev]

Some more composite terms n (may be not in order):

64*2^26 + 1 = 4294967297,
128*2^57 + 1 = 18446744073709551617,
256*2^120 + 1 = 340282366920938463463374607431768211457,
512*2^247 + 1,
1024*2^502 + 1,
2048*2^1013 + 1,
4096*2^2036 + 1,
8192*2^4083 + 1

Here the first factor in each expression for n is the multiplicative
order of 2 modulo n (in other words, the period on 1/n in binary).
Quite interesting, here they all are powers of 2 -- this does not hold
for smaller terms however -- e.g., 12801 = 200*2^6 + 1.

Regards,
Max


On Sun, May 25, 2014 at 9:04 AM, Tw Mike <mt.kongtong at gmail.com> wrote:
> Dear seqfans,
>
> OEIS A226181:
>
> 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79,
> ...
> Primes p such that p−1 divided by the period of the binary expansion of 1/p
> equals 2^x for some nonnegative integer x.
>
> Composite numbers matching the conditions below 108 are:
> 12801, 348161, 3225601
>
> Maybe all composite numbers matching the conditions are both Poulet numbers
> and Proth numbers, but Joni Teräväinen say suspect that there is an n such
> that n satisfies the condition but is not a Proth number. So more composite
> numbers for A226181 are needed.
>
> See the related question at mathoverflow: Are all counterexamples of OEIS
> A226181 both Poulet numbers and Proth numbers? taged by nt.number-theory,
> prime-numbers and sequences-and-series
>
> Yours mike,
>
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