A385658 - OEIS

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.

1, 2, 5, 17, 59, 19, 43, 17, 19, 89, 47, 67, 257, 89, 173, 11, 103, 67, 103, 191, 29, 89, 101, 139, 19, 13, 19, 79, 79, 271, 223, 149, 131, 5, 37, 31, 593, 149, 353, 109, 293, 293, 17, 19, 97, 83, 59, 79, 883, 101, 71, 13, 199, 113, 1013, 29, 1279, 7, 181, 383, 269, 197, 17

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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).

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..500

EXAMPLE

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.

MATHEMATICA

Tau[n_]:=Tau[n]=RamanujanTau[n];

P[n_, x_]:=P[n, x]=Sum[Tau[k]x^(n-k), {k, 1, n}];

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]

PROG

(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

CROSSREFS

Cf. A000040, A000594, A385676, A385678.

Sequence in context: A134128 A119254 A278394 * A056679 A084528 A149993

Adjacent sequences: A385655 A385656 A385657 * A385659 A385660 A385661

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 03 2025

STATUS

approved