cp's OEIS Frontend

A385658 Least prime p < 2n*(n+1) such that the polynomial Sum_{k=1..n} tau(k)*x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where tau is Ramanujan's tau function given by A000594.

%I A385658 #30 Aug 11 2025 00:13:24
%S A385658 1,2,5,17,59,19,43,17,19,89,47,67,257,89,173,11,103,67,103,191,29,89,
%T A385658 101,139,19,13,19,79,79,271,223,149,131,5,37,31,593,149,353,109,293,
%U A385658 293,17,19,97,83,59,79,883,101,71,13,199,113,1013,29,1279,7,181,383,269,197,17
%N A385658 Least prime p < 2n*(n+1) such that the polynomial Sum_{k=1..n} tau(k)*x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where tau is Ramanujan's tau function given by A000594.
%C A385658 Conjecture: a(n) > 1 for all n > 1. In other words, for each n = 2,3,... the polynomial x^(n-1) + tau(2)*x^(n-2) + ... + tau(n) is irreducible modulo some prime p < 2n*(n+1).
%H A385658 Zhi-Wei Sun, <a href="/A385658/b385658.txt">Table of n, a(n) for n = 1..500</a>
%e A385658 a(5) = 59 since the prime 59 is smaller than 2*5*(5+1) = 60, and 59 is the least prime p such that the polynomial tau(1)*x^4 + tau(2)*x^3 + tau(3)*x^2 + tau(4)*x + tau(5) is irreducible modulo p.
%t A385658 Tau[n_]:=Tau[n]=RamanujanTau[n];
%t A385658 P[n_,x_]:=P[n,x]=Sum[Tau[k]x^(n-k),{k,1,n}];
%t A385658 tab={};Do[Do[If[IrreduciblePolynomialQ[P[n, x], Modulus->Prime[k]]==True, tab=Append[tab,Prime[k]]; Goto[aa]], {k, 1, PrimePi[2n(n+1)-1]}];tab=Append[tab,1]; Label[aa]; Continue, {n,1,63}];Print[tab]
%o A385658 (PARI) a(n) = forprime(p=2, 2*n*(n+1)-1, if (polisirreducible(Mod(sum(k=1, n, ramanujantau(k)*x^(n-k)), p)), return(p))); 1; \\ _Michel Marcus_, Aug 04 2025
%Y A385658 Cf. A000040, A000594, A385676, A385678.
%K A385658 nonn
%O A385658 1,2
%A A385658 _Zhi-Wei Sun_, Aug 03 2025