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A385678 Least prime p <= n^2 - 2*n + 4 such that the polynomial Sum_{k=1..n} phi(k)*x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where phi is Euler's totient function given by A000010.

%I A385678 #20 Aug 12 2025 00:01:58
%S A385678 1,2,3,1,1,7,31,13,67,7,67,13,53,7,11,19,101,239,37,23,13,103,263,89,
%T A385678 79,29,47,23,167,317,139,73,283,7,223,71,83,29,1117,503,83,167,811,
%U A385678 349,17,3,263,37,157,317,11,7,43,283,17,79,193,293,257,233
%N A385678 Least prime p <= n^2 - 2*n + 4 such that the polynomial Sum_{k=1..n} phi(k)*x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where phi is Euler's totient function given by A000010.
%C A385678 Conjecture: a(n) > 1 for all n > 5.
%C A385678 Note that Sum_{k=1..4} phi(k)*x^(4-k) = (x + 1)*(x^2 + 2) and Sum_{k=1..5} phi(k)*x^(5-k) = (x^2 - x + 2)*(x^2 + 2*x + 2).
%C A385678 See also A385658 and A385676 for similar conjectures.
%H A385678 Zhi-Wei Sun, <a href="/A385678/b385678.txt">Table of n, a(n) for n = 1..400</a>
%F A385678 a(9) = 67 since 67 = 9^2 - 2*9 + 4 is the least prime p such that the polynomial Sum_{k=1..9}phi(k)*x^(9-k) is irreducible modulo p.
%t A385678 P[n_, x_]:=P[n, x]=Sum[EulerPhi[k]*x^(n-k), {k, 1, n}];
%t A385678 tab={};Do[Do[If[IrreduciblePolynomialQ[P[n, x], Modulus->Prime[k]]==True, tab=Append[tab,Prime[k]]; Goto[aa]], {k, 1, PrimePi[n^2-2n+4]}];
%t A385678 tab=Append[tab,1]; Label[aa]; Continue, {n, 1, 60}];Print[tab]
%o A385678 (PARI) a(n) = forprime(p=2, n^2 - 2*n + 4, if (polisirreducible(Mod(sum(k=1, n, eulerphi(k)*x^(n-k)), p)), return(p))); 1; \\ _Michel Marcus_, Aug 04 2025
%Y A385678 Cf. A000010, A000040, A385658, A385676.
%K A385678 nonn
%O A385678 1,2
%A A385678 _Zhi-Wei Sun_, Aug 04 2025