A385678 - OEIS

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.

1, 2, 3, 1, 1, 7, 31, 13, 67, 7, 67, 13, 53, 7, 11, 19, 101, 239, 37, 23, 13, 103, 263, 89, 79, 29, 47, 23, 167, 317, 139, 73, 283, 7, 223, 71, 83, 29, 1117, 503, 83, 167, 811, 349, 17, 3, 263, 37, 157, 317, 11, 7, 43, 283, 17, 79, 193, 293, 257, 233

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OFFSET

1,2

COMMENTS

Conjecture: a(n) > 1 for all n > 5.

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).

See also A385658 and A385676 for similar conjectures.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..400

FORMULA

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.

MATHEMATICA

P[n_, x_]:=P[n, x]=Sum[EulerPhi[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[n^2-2n+4]}];

tab=Append[tab, 1]; Label[aa]; Continue, {n, 1, 60}]; Print[tab]

PROG

(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

CROSSREFS

Cf. A000010, A000040, A385658, A385676.

Sequence in context: A217897 A135900 A338072 * A173272 A326303 A047789

Adjacent sequences: A385675 A385676 A385677 * A385679 A385680 A385681

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 04 2025

STATUS

approved