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A348059 a(n) = Min {k > n : A235137(k) == -n (mod k)}, or -1 if no such minimum exists.

%I A348059 #27 Nov 04 2021 22:16:17
%S A348059 2,4,9,8,6,180,10,15,14,12,121,35,22,20,18,32,289,28,34,24,38,57,529,
%T A348059 45,46,44,81,40,546,36,58,64,62,93,50,56,210,68,74,48,42,76,82,123,54,
%U A348059 129,510,117,94,92,3376098,88,690,329,106,80,2898,265,70,72,66
%N A348059 a(n) = Min {k > n : A235137(k) == -n (mod k)}, or -1 if no such minimum exists.
%C A348059 Conjecture: For all n, a(n) > 0.
%t A348059 Giuga1[mu_][n_] := Giuga1[mu][n] =
%t A348059 Mod[Sum[PowerMod[i, EulerPhi[n], n], {i, 1, n}] + mu, n] == 0;
%t A348059 A348059[n_] := A348059[n] =
%t A348059 {Clear[ww];Do[If[Giuga1[n][i], ww = i; Break[]], {i, n + 1, 20000000}]; ww} [[1]];
%t A348059 Table[A348059[n],{n,49}]
%o A348059 (PARI) a(n) = my(k=n+1); while (sum(i=1, k , Mod(i, k)^eulerphi(k)) != -n, k++); k; \\ _Michel Marcus_, Sep 28 2021
%Y A348059 Cf. A235137, A348058.
%K A348059 nonn
%O A348059 1,1
%A A348059 _José María Grau Ribas_, Sep 26 2021