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a(n) is the smallest k > 1 such that gcd(k, n^k - n) = 1, for n > 1.
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%I #52 Aug 13 2018 15:33:14

%S 35,35,77,77,143,55,55,77,119,119,35,55,187,143,77,35,35,77,143,247,

%T 95,35,77,77,77,55,55,143,77,77,35,35,247,143,143,35,35,77,77,143,55,

%U 95,119,119,77,35,55,143,143,77,35,35,119,299,221,55,35,77,77,77,55,55,187,119

%N a(n) is the smallest k > 1 such that gcd(k, n^k - n) = 1, for n > 1.

%C Conjecture: all the terms are in A121707. If k is a term, then k is an "anti-Carmichael number": p-1 does not divide k-1 for every prime p dividing k.

%C The sequence is unbounded, since a(m!) > m.

%C Prediction: a(n) < n for all sufficiently large n.

%C GCD(n, a(n)) = 1. a(n) is odd. Is a(n) squarefree? - _David A. Corneth_, Aug 13 2018

%o (PARI) a(n) = {my(k=2); while (gcd(k, n^k - n) != 1, k++); k;} \\ _Michel Marcus_, Aug 13 2018

%o (PARI) a(n) = {my(k=3); while (gcd(k, n^k - n) != 1, k+=2; while(gcd(n, k) > 1, k+=2)); k} \\ _David A. Corneth_, Aug 13 2018

%Y Cf. A121707, A267999.

%K nonn

%O 2,1

%A _Thomas Ordowski_, Aug 13 2018

%E More terms from _Michel Marcus_, Aug 13 2018