A385678 -id:A385678 - 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.

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

Offset: 1

Zhi-Wei Sun, Aug 03 2025

nonn

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).

			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.

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

Cf. A000040, A000594, A385676, A385678.

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]

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

A385676 Least prime p <= 2n^2 - n + 1 such that the polynomial Sum_{k=1..n} sigma(k) x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where sigma is given by A000203.

Original entry on oeis.org

1, 2, 3, 2, 1, 5, 11, 29, 2, 47, 5, 31, 13, 379, 37, 251, 23, 29, 67, 97, 41, 131, 11, 173, 41, 139, 79, 103, 281, 19, 7, 53, 71, 281, 131, 19, 3, 43, 149, 23, 347, 47, 29, 107, 107, 47, 823, 47, 311, 547, 67, 419, 263, 379, 349, 23, 227, 349, 19, 113

Offset: 1

Zhi-Wei Sun, Aug 04 2025

nonn

Conjecture: a(n) > 1 except for n = 1, 5.
Note that Sum_{k=1..5} sigma(k) * x^(5-k) = x^4 + 3*x^3 + 4*x^2 + 7*x + 6 = (x + 2)*(x^3 + x^2 + 2*x + 3).
See A385678 for a similar conjecture involving Euler's totient function.

			a(14) = 379 since 379 = 2*14^2 - 14 + 1 is the least prime p such that Sum_{k=1..14} sigma(k) * x^(14-k) is irreducible modulo p.

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

Cf. A000040, A000203, A385658, A385678.

sigma[n_]:=sigma[n]=DivisorSigma[1,n];
P[n_, x_]:=P[n, x]=Sum[sigma[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^2-n+1]}];
tab=Append[tab,1]; Label[aa]; Continue, {n, 1, 60}];Print[tab]

a(n) = forprime(p=2, 2*n^2 - n + 1, if (polisirreducible(Mod(sum(k=1, n, sigma(k)*x^(n-k)), p)), return(p))); 1; \\ Michel Marcus, Aug 04 2025

A386827 Least prime n < p < 2n(n-1) such that the polynomial Sum_{k=1..n} x^(n-k)/k is irreducible modulo p, or 1 if such a prime p does not exist.

Original entry on oeis.org

1, 3, 7, 13, 7, 11, 83, 11, 43, 103, 41, 29, 89, 67, 43, 23, 41, 67, 131, 269, 47, 151, 43, 149, 191, 127, 29, 113, 263, 173, 61, 463, 223, 67, 61, 127, 103, 97, 47, 271, 89, 59, 337, 281, 157, 541, 269, 277, 73, 337, 463, 379, 223, 1481, 827, 797, 397, 101, 337, 431

Offset: 1

Zhi-Wei Sun, Aug 04 2025

nonn

Conjecture: a(n) > 1 for all n > 1. In other words, for each n = 2,3,... there is a prime p with n < p < 2*n*(n-1) such that the polynomial Sum_{k=1..n} x^(n-k)/k is irreducible modulo p.

			a(7) = 83 since 83 = 2*7*(7-1) - 1 is the least prime p > 7 such that the polynomial x^6 + x^5/2 + x^4/3 + x^3/4 + x^2/5 + x/6 + 1/7 is irreducible modulo p.

Zhi-Wei Sun, Table of n, a(n) for n = 1..400
Zhi-Wei Sun, On the polynomial x^{n-1}+x^{n-2}/2+...+1/n, Question 498716 in MathOverflow, August 5, 2025.

Cf. A000040, A385658, A385676, A385678, A386828, A386850.

P[n_, x_]:=P[n, x]=Sum[x^(n-k)/k, {k, 1, n}];
tab={};Do[Do[If[IrreduciblePolynomialQ[P[n, x], Modulus->Prime[k]]==True, tab=Append[tab,Prime[k]]; Goto[aa]], {k, PrimePi[n]+1, PrimePi[2n(n-1)-1]}];
tab=Append[tab,1]; Label[aa]; Continue, {n,1,60}];Print[tab]

a(n) = forprime(p=n+1, 2*n*(n-1)-1, if (polisirreducible(Mod(sum(k=1, n, x^(n-k)/k), p)), return(p))); 1; \\ Michel Marcus, Aug 05 2025

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.

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A385676 Least prime p <= 2n^2 - n + 1 such that the polynomial Sum_{k=1..n} sigma(k) x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where sigma is given by A000203.

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A386827 Least prime n < p < 2n(n-1) such that the polynomial Sum_{k=1..n} x^(n-k)/k is irreducible modulo p, or 1 if such a prime p does not exist.

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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.

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A385676 Least prime p <= 2*n^2 - n + 1 such that the polynomial Sum_{k=1..n} sigma(k) * x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where sigma is given by A000203.

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A386827 Least prime n < p < 2*n*(n-1) such that the polynomial Sum_{k=1..n} x^(n-k)/k is irreducible modulo p, or 1 if such a prime p does not exist.

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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.

A385676 Least prime p <= 2n^2 - n + 1 such that the polynomial Sum_{k=1..n} sigma(k) x^(n-k) is irreducible modulo p, or 1 if such p does not exist, where sigma is given by A000203.

A386827 Least prime n < p < 2n(n-1) such that the polynomial Sum_{k=1..n} x^(n-k)/k is irreducible modulo p, or 1 if such a prime p does not exist.