A385658 -id:A385658 - cp's OEIS frontend

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.

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

A385678 Least prime p <= n^2 - 2n + 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.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Aug 04 2025

nonn

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.

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

Cf. A000010, A000040, A385658, A385676.

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]

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

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.

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

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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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A385678 Least prime p <= n^2 - 2n + 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.

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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

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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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.

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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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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.

A385678 Least prime p <= n^2 - 2n + 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.

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.