[seqfan] Almost A111076

Tomasz Ordowski tomaszordowski at gmail.com
Sun Nov 10 11:28:08 CET 2019


Hello SeqFans!

Let a(n) be the smallest natural base k such that
k^m == 1 (mod n) and k^{m/2} =/= 1 (mod n),
where m = lambda(n) = A002322(n).

a(n) is the smallest positive integer k such that
gcd(k,n) = 1 and k^{lambda(n)/2} =/= 1 (mod n),
where lambda(n) is the Carmichael function.

2, 3, 2, 3, 2, 5, 3, 3, 2, 3, 2, 5, 2, 3, 2, 3, 3, 5, 2, 3,
2, 7, 5, 5, 2, 5, 2, 3, 2, 7, 3, 3, 2, 3, 2, 5, 2, 3, 2, 3,
3, 5, 2, 3, 2, 5, 5, 5, 3, 3, 5, 5, 2, 5, 2, 3, 2, 3, 2, 7,
2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 7, 5, 5, 5, 2, 3, 2, 5, 3, 3,
2, 3, 2, 5, 3, 3, 2, 3, 3, 7, 2, 3, 2, 5, 2, 5, 5, 3, 2, 3,
...
It should be noted that, for n > 2, a(n) <= A111076(n);
a(n) < A111076(n) for n = 26, 41, 43, 52, 78, 82, 93, ...

Conjecture: a(n) is a prime for all n.
Checked up to n = 10^8 by Amiram Eldar.
I am asking for a proof or a counterexample.

Best regards,

Thomas Ordowski
_____________________
Cf. A111076 and A229708.



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