[seqfan] Re: Almost Carmichael numbers

[Tomasz Ordowski]

Dear David,

You forgot about the condition omega(n) > 2 and mistyped p-1 / n-1 instead
of p-1 | n-1.
The given definition does not exclude odd semiprimes n = pq with primes p <
q such that p-1 | n-1, equivalently p-1 | q-1. It's also interesting
numbers.

Thomas

wt., 18 cze 2019 o 23:42 David Sycamore via SeqFan <seqfan at list.seqfan.eu>
napisał(a):

> Dear Seqfans,
>
> Odd square-free composite numbers n, for which omega(n)=k, and n has
> exactly k-1 prime factors p, such that p-1/n-1:
>
> 105, 165, 231, 285, 345, 385, 465, 645, 705, 805, 885, 1005, 1045, 1065,
> 1185, 1221, 1245, 1545, 1551, 1605, 1645, 1653, 1771....
>
> The first 68 terms all have o(n)=3; 5005 is the first term with o(n)=4.
>
> Sequence of smallest such numbers having k prime divisors; k>=3 :
>
> 105, 5005, 185185, 8964865....
>
> (Have not found any higher terms than these...).
>
> Comments and more data welcome
>
> David.
>
> NB: Sequence of odd squarefree composite numbers n for which only the
> least and greatest prime divisors have the property p-1/n-1 seems to be:
> 231, 1045, 1653, 4371, 4641, 5365, 6545, 10011, 10857... (draft terms, I
> have no code for this yet).  These numbers seem to be much rarer than the
> first sequence above, appearing (based on this small sample) at around  the
> same rate as the Carmichael numbers.
>
>
>
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