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A319484 a(n) is the smallest k > 1 such that n^k == n (mod k) and gcd(k, b^k-b) = 1 for some b <> n.

%I A319484 #45 Dec 25 2018 17:28:43
%S A319484 35,35,7957,16531,1247,4495,35,817,2501,697,55,55,143,221,35,35,1247,
%T A319484 493,221,95,35,35,77,253,115,403,247,247,203,35,155,155,697,187,35,35,
%U A319484 35,589,221,95,533,35,287,77,55,55,115,221,329,35,35,221,221,689,55,35,35
%N A319484 a(n) is the smallest k > 1 such that n^k == n (mod k) and gcd(k, b^k-b) = 1 for some b <> n.
%C A319484 a(n) is the smallest k > 1 such that n^k == n (mod k) and p-1 does not divide k-1 for every prime p dividing k, see A121707.
%C A319484 We have A000790(n) < a(n) <= A316940(n) for n > 0.
%C A319484 It seems that the sequence is unbounded like A316940.
%C A319484 The term a(5) = 4495 = 5*29*31 is not semiprime.
%e A319484 a(6) = 35 since 6^35 == 6 (mod 35) and 35 = 5*7 is the smallest "anti-Carmichael number": 5-1 does not divide 7-1. We have gcd(35,2^35-2) = 1.
%o A319484 (PARI) isac(n) = {my(f = factor(n)[,1]); for (i=1, #f, if (((n-1) % (f[i]-1)) == 0, return (0));); return (1);}
%o A319484 isok(n,k) = {if (Mod(n, k)^k != Mod(n, k), return (0)); return (isac(k));}
%o A319484 a(n) = {my(k=2); while (!isok(n,k), k++); return (k);} \\ _Michel Marcus_, Oct 27 2018
%Y A319484 Cf. A000790, A121707, A316940.
%K A319484 nonn
%O A319484 0,1
%A A319484 _Thomas Ordowski_, Oct 26 2018
%E A319484 More terms from _Michel Marcus_, Oct 26 2018