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A348058 a(n) = Min {k > n : A235137(k) == n (mod k)}, or -1 if no such minimum exists.

%I A348058 #54 Jun 20 2025 17:36:14
%S A348058 2,3,10,5,14,7,15,16,22,11,21,13,114,156,34,17,38,19,33,25,45,23,35,
%T A348058 80,186,228,58,29,30,31,51,64,63,76,57,37,258,2244,55,41,86,43,69,104,
%U A348058 94,47,65,160,1518,372,106,53,354,81,87,624,99,59,77,61,402
%N A348058 a(n) = Min {k > n : A235137(k) == n (mod k)}, or -1 if no such minimum exists.
%C A348058 Conjecture: For all n, a(n) > 0.
%C A348058 If a(673) > 0 then a(673) > 10^10.
%H A348058 José María Grau Ribas, <a href="/A348058/b348058.txt">Table of n, a(n) for n = 1..672</a>
%t A348058 Giuga1[mu_][n_] := Giuga1[mu][n] =
%t A348058 Mod[Sum[PowerMod[i, EulerPhi[n], n], {i, 1, n}] - mu, n] == 0;
%t A348058 A348058[n_] := A348058[n] =
%t A348058 {Clear[ww];  Do[If[Giuga1[n][i], ww = i; Break[]], {i, n + 1, 20000000}]; ww}[[1]];
%t A348058 Table[A348058[n],{n,61}]
%o A348058 (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 A348058 Cf. A235137, A348059.
%K A348058 nonn
%O A348058 1,1
%A A348058 _José María Grau Ribas_, Sep 26 2021