[seqfan] Almost Carmichael numbers
David Sycamore
djsycamore at yahoo.co.uk
Tue Jun 18 13:30:59 CEST 2019
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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