Zhi-Wei Sun - cp's OEIS frontend

A386850 Least prime n < p <= (n-1)*(2n-1) such that Sum_{k=1..n} x^(n-k)/k! is irreducible modulo p, or 1 if such a prime p does not exist.

1, 3, 7, 7, 13, 41, 13, 31, 29, 31, 37, 23, 97, 331, 53, 101, 47, 89, 43, 199, 53, 43, 47, 107, 83, 61, 149, 37, 353, 127, 113, 199, 173, 107, 67, 401, 349, 101, 347, 47, 79, 89, 83, 241, 139, 641, 673, 103, 491, 179, 383, 293, 61, 439, 397, 547, 79, 1301, 379, 277

Offset: 1

Zhi-Wei Sun, Aug 05 2025

nonn

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

Note that Sum_{k>0}x^k/k! = e^x - 1.

			a(2) = 3 since 3 is the only prime in the interval (2, (2-1)*(2*2-1)] and x + 1/2 is irreducible modulo 3.

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

Cf. A000040, A000142, A386827, A386828.

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[(n-1)(2n-1)]}];tab=Append[tab,1]; Label[aa]; Continue, {n, 1, 60}];Print[tab]

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

A386828 Least prime n < p <= 2*n^2 + 1 such that the polynomial Sum_{k=1..n} x^(n-k)/k^2 is irreducible modulo p, or 1 if such a prime p does not exist.

Original entry on oeis.org

1, 3, 19, 13, 7, 17, 19, 13, 17, 13, 17, 31, 139, 151, 19, 181, 113, 157, 79, 89, 89, 71, 37, 31, 197, 31, 199, 149, 83, 37, 127, 59, 647, 89, 47, 47, 157, 197, 97, 79, 601, 59, 79, 67, 71, 487, 223, 577, 359, 83, 269, 269, 251, 461, 229, 67, 1777, 859, 1091, 701

Offset: 1

Zhi-Wei Sun, Aug 05 2025

nonn

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

We also have similar conjectures for Sum_{k=1..n} x^(n-k)/k^s with other values of s.

			a(3) = 19 since 19 = 2*3^2 + 1 is the least prime p > 3 such that the polynomial x^2 + x/2 + 1/3 is irreducible modulo p.

Zhi-Wei Sun, Table of n, a(n) for n = 1..300

Cf. A000040, A386827.

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

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

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.

Original entry on oeis.org

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

A386703 The residue of p(n) modulo q(n) in the interval (-q(n)/2, q(n)/2], where p(n) = A000041(n) and q(n) = A000009(n).

Original entry on oeis.org

0, 0, 1, 1, 1, -1, 0, -2, -2, 2, -4, 2, -7, 3, -13, 7, -7, 17, 4, -13, 32, 23, 7, -11, -30, -39, -62, -56, -43, -20, 42, 159, -161, 22, 258, -59, 357, 95, -239, -504, 483, 412, 471, 719, -978, -426, 434, -1137, 533, -622, -1780, 2087, 2251, -2669, -1562, 831, -3372, 1772

Offset: 1

Zhi-Wei Sun, Jul 30 2025

sign

Conjecture: |a(n)| > 1 for all n > 7.
This has been verified for all n = 8..10^5.
Verified for all n <= 2000000. - Vaclav Kotesovec, Jul 30 2025

			a(6) = -1  since p(6) = 11 is congruent to -1 modulo q(6) = 4.
a(7) = 0 since p(7) = 15 is congruent to 0 modulo q(7) = 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, A conjecture involving the partition function and the strict partition function, Question 498447 at MathOverflow, July 30, 2025.

Cf. A000009, A000041.

rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,(1-n)/2];
a[n_]=rMod[PartitionsP[n],PartitionsQ[n]];Table[a[n],{n,1,70}]

A383274 a(n) = Sum_{i,j = 0..n} C(n, i)^2C(n, j)^2C(i+j, i)*2^(i+j).

Original entry on oeis.org

1, 13, 441, 20629, 1119361, 66116013, 4126228569, 267666251733, 17868312820737, 1219477111897933, 84701899713767161, 5967906378862013973, 425503428034568158081, 30642774518964618986989, 2225692868157573335052441, 162858794856607965831417429, 11993850186156155815298686977

Offset: 0

Zhi-Wei Sun, Apr 26 2025

nonn

Conjecture 1: Let S(p) = Sum_{0

If p == 1,9 (mod 20) with p = x^2 + 5*y^2, then S(p) == 4*x^2-2*p (mod p^2).

If p == 3,7 (mod 20) with 2*p = x^2 + 5*y^2, then S(p) == 2*x^2 - 2*p (mod p^2).

If p == 11, 13, 17, 19 (mod 20), then S(p) == 0 (mod p^2).

Conjecture 2: For any prime p == 1,3,7,9 (mod 20), we have Sum_{0

Both conjectures have been verified for odd primes smaller than 1000.

			a(1) = 13 since Sum_{i,j = 0,1}C(1,i)^2*C(1,j)^2*C(i+j,i)^2*2^(i+j) = Sum_{i,j = 0,1} C(i+j,i)^2*2^(i+j) = 2^0 + 2^1 + 2^1 + C(2,1)*2^2 = 13.

Zhi-Wei Sun, Table of n, a(n) for n = 0..85
Zhi-Wei Sun, On a_n(x) = Sum_{i,j=0..n}C(n,i)^2*C(n,j)^2*C(i+j,i)*x^(i+j) (I), Question 491655 at MathOverflow, April 24, 2025.
Zhi-Wei Sun, A family of polynomials and related congruences and series, arXiv:2505.02767 [math.NT], 2025.

Cf. A005259.

a := proc(n) option remember; local i, j; add(add(binomial(n, i)^2 * binomial(n, j)^2 * binomial(i+j, i) * 2^(i+j), i = 0..n), j = 0..n) end: seq(a(n), n=0..16);  # Peter Luschny, Apr 27 2025

a[n_] := a[n] = Sum[Binomial[n,i]^2*Binomial[n,j]^2*Binomial[i+j,i]*2^(i+j), {i,0,n}, {j,0,n}]; Table[a[n], {n,0,17}]

from math import comb
def A383274(n): return sum((comb(n,i)**2<Chai Wah Wu, Apr 27 2025

a(n) ~ 3^(4*n+3) / (8 * sqrt(7) * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 27 2025

A379654 Least positive integer k <= n such that |tau(k)|*n + 1 is prime, or 0 if such k does not exist, where the tau function is given by A000594.

Original entry on oeis.org

1, 1, 2, 1, 5, 1, 5, 2, 3, 1, 4, 1, 2, 2, 11, 1, 2, 1, 2, 5, 13, 1, 4, 2, 2, 3, 5, 1, 3, 1, 5, 2, 3, 6, 3, 1, 21, 15, 2, 1, 3, 1, 2, 11, 5, 1, 2, 2, 6, 2, 3, 1, 4, 2, 2, 11, 7, 1, 3, 1, 3, 2, 3, 5, 3, 1, 2, 3, 2, 1, 4, 1, 2, 2, 2, 6, 10, 1, 15, 3, 4, 1, 2, 2, 5, 3, 2, 1, 2, 2, 6, 12, 4, 3, 2, 1, 7, 3, 2, 1

Offset: 1

Zhi-Wei Sun, Dec 28 2024

nonn

Conjecture 1: a(n) > 0 for all n > 0. In other words, for each positive integer n, there is a number k among 1,...,n such that |tau(k)|*n + 1 is prime.
Conjecture 2: For each integer n > 1 not equal to 22, there is a number k among 1,...,n such that |tau(k)|*n - 1 is prime.
We have verified both conjectures for n up to 10^8.

			a(1) = 1 since 1*|tau(1)| + 1 = 2 is a prime.
a(5) = 5 since 5*|tau(5)| + 1 = 5*4830 + 1 = 24151 is prime, and 5*|tau(k)| + 1 is composite for every k = 1, 2, 3, 4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A000594, A034693.

t[n_]:=t[n]=Abs[RamanujanTau[n]];
L={};Do[Do[If[PrimeQ[t[k]n+1],L=Append[L,k];Goto[aa]],{k,1,n}];L=Append[L,0];Label[aa],{n,1,100}];Print[L]

A376336 Least positive integer m such that tau(m) is divisible by n, where the Ramanujan tau function is given by A000594.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 3, 2, 3, 5, 8, 2, 7, 3, 5, 4, 239, 3, 89, 8, 3, 8, 4, 2, 25, 7, 6, 3, 13, 5, 139, 4, 8, 239, 5, 3, 191, 89, 11, 8, 257, 3, 19, 8, 10, 4, 67, 6, 15, 40, 239, 7, 107, 6, 8, 6, 89, 13, 61, 8, 9, 139, 3, 4, 35, 8, 31, 239, 5, 5, 137, 6, 2069, 191, 40, 178, 19, 11, 25, 8, 9, 257, 227, 3, 239, 19, 26, 8, 59, 10, 7, 4, 278, 67, 89, 6, 863, 15, 24, 40

Offset: 1

Zhi-Wei Sun, Dec 22 2024

nonn

Conjecture: a(n) exists for any positive integer n. Moreover, a(n) <= n^2 for all n > 0.

It seems that for some primes p the value of a(p) is relatively large. For example, 4327 is a prime with a(4327) = 316717, 9133 is a prime with a(9133) = 789977, and 9643 is a prime with a(9643) = 1001401.

			a(5) = 5 since tau(5) = 4830 is divisible by 5, but none of tau(1) = 1, tau(2) = -24, tau(3) = 252 and tau(4) = -1472 is a multiple of 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000594.

f[n_]:=f[n]=RamanujanTau[n]; L={}; Do[m=1;Label[bb];If[Mod[f[m],n]==0,L=Append[L,m];Goto[aa]];m=m+1;Goto[bb];Label[aa],{n,1,100}];Print[L]
(* Or: *)
a[n_] := SelectFirst[Range[1, 30000], Divisible[RamanujanTau[#], n] &]; Array[a, 1000] (* Peter Luschny, Dec 22 2024 *)

first(n) = {
	my(todo = [1..n], res = vector(n, i, oo));
	for(i = 1, oo,
		c = ramanujantau(i);
		for(j = 1, #todo,
			if(res[todo[j]] > i && c % todo[j] == 0,
				res[todo[j]] = i;
				todo = setminus(todo, Set(todo[j]));
				if(#todo == 0,
					return(res)
				)
			)
		);
);} \\ David A. Corneth, Dec 23 2024

from itertools import count
tau = delta_qexp(30000)  # adjust search length for n > 1000
a = lambda n: next((k for k in count(1) if n.divides(tau[k])))
print([a(n) for n in srange(1, 1001)])  # Peter Luschny, Dec 22 2024

cp's OEIS Frontend

User: Zhi-Wei Sun

A386850 Least prime n < p <= (n-1)*(2n-1) such that 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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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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A386828 Least prime n < p <= 2*n^2 + 1 such that the polynomial Sum_{k=1..n} x^(n-k)/k^2 is irreducible modulo p, or 1 if such a prime p does not exist.

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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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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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A386703 The residue of p(n) modulo q(n) in the interval (-q(n)/2, q(n)/2], where p(n) = A000041(n) and q(n) = A000009(n).

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A383274 a(n) = Sum_{i,j = 0..n} C(n, i)^2*C(n, j)^2*C(i+j, i)*2^(i+j).

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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.

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.

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.

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.

A383274 a(n) = Sum_{i,j = 0..n} C(n, i)^2C(n, j)^2C(i+j, i)*2^(i+j).