A242213 -id:A242213 - cp's OEIS frontend

A242210 Number of primes p < prime(n) such that the Bernoulli number B_{p-1} is a primitive root modulo prime(n).

0, 0, 1, 2, 1, 2, 2, 1, 4, 2, 3, 6, 3, 2, 5, 6, 5, 7, 4, 6, 6, 10, 11, 12, 8, 10, 9, 12, 10, 13, 9, 9, 10, 10, 17, 11, 7, 11, 18, 22, 15, 11, 12, 15, 21, 15, 10, 15, 23, 18, 26, 15, 15, 22, 26, 22, 25, 19, 26, 22, 22, 20, 17, 23, 20, 28, 17, 18, 28, 22

Offset: 1

Zhi-Wei Sun, May 07 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 2. In other words, for any prime p > 3, there exists a prime q < p such that the Bernoulli number B_{q-1} is a primitive root modulo p.
(ii) For any prime p > 13, there exists a prime q < p such that the Euler number E_{q-1} is a primitive root modulo p.
We have verified part (i) for n up to 4.2*10^5, and part (ii) for primes p below 10^6.

			a(4) = 2 since 3 is a primitive root modulo prime(4) = 7, and both B_{2-1} = - 1/2 and B_{5-1} = - 1/30 are congruent to 3 modulo 7.
a(5) = 1 since B_{3-1} = 1/6 == 2 (mod 11) with 2 a primitive root modulo prime(5) = 11.
a(8) = 1 since B_{17-1} = -3617/510 == -4 (mod 19) with -4 a primitive root modulo prime(8) = 19.

Zhi-Wei Sun, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, Notes on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A027641, A027642, A122045, A242193, A242194, A242213.

rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
f[k_]:=BernoulliB[Prime[k]-1]
dv[n_]:=Divisors[n]
Do[m=0;Do[If[rMod[f[k],Prime[n]]==0,Goto[aa]];Do[If[rMod[f[k]^(Part[dv[Prime[n]-1],i])-1,Prime[n]]==0,Goto[aa]],{i,1,Length[dv[Prime[n]-1]]-1}];m=m+1;Label[aa];Continue,{k,1,n-1}];Print[n," ",m];Continue,{n,1,70}]

A242222 Number of primes p <= (prime(n)+1)/2 such that the harmonic number H(p-1) = sum_{0

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 1, 1, 1, 2, 3, 4, 4, 5, 6, 3, 2, 3, 2, 3, 2, 6, 6, 4, 6, 4, 8, 7, 9, 5, 7, 11, 5, 11, 5, 6, 6, 11, 8, 12, 7, 8, 9, 8, 11, 7, 13, 18, 8, 18, 14, 8, 9, 14, 18, 17, 7, 14, 11, 9, 19, 10, 12, 7, 21, 5, 15, 19, 15

Offset: 1

Zhi-Wei Sun, May 08 2014

nonn

Conjecture: a(n) > 0 for all n > 3. In other words, for any prime p > 5, there exists a prime q <= (p+1)/2 such that the harmonic number H(q-1) = sum_{0

			a(6) = 1 since 7 is a prime not exceeding (prime(6)+1)/2 = 7, and H(7-1) = 49/20 == -6 (mod 13) with -6 a primitive root modulo prime(6) = 13.
a(8) = 1 since 5 is a prime not exceeding (prime(8)+1)/2 = 10, and H(5-1) = 25/12 == -9 (mod 19) with -9 a primitive root modulo prime(8) = 19.
a(9) = 1 since 11 is a prime not exceeding (prime(9)+1)/2 = 12, and H(11-1) = 7381/2520 == -9 (mod 23) with -9 a primitive root modulo prime(9) = 23.

Zhi-Wei Sun, Table of n, a(n) for n = 1..700
Zhi-Wei Sun, Notes on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A001008, A002805, A242210, A242213, A242223.

rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
f[k_]:=HarmonicNumber[Prime[k]-1]
dv[n_]:=Divisors[n]
Do[m=0;Do[If[rMod[f[k],Prime[n]]==0,Goto[aa]];Do[If[rMod[f[k]^(Part[dv[Prime[n]-1],i])-1,Prime[n]]==0,Goto[aa]],{i,1,Length[dv[Prime[n]-1]]-1}];m=m+1;Label[aa];Continue,{k,1,PrimePi[(Prime[n]+1)/2]}];Print[n," ",m];Continue,{n,1,70}]

A242241 Least prime p such that H(2,n) = sum_{k=1..n}1/k^2 == 0 (mod p) but there is no 0 < k < n with H(2,k) == 0 (mod p), or 1 if such a prime p does not exist.

Original entry on oeis.org

1, 5, 7, 41, 11, 13, 266681, 17, 19, 178939, 23, 18500393, 40799043101, 29, 31, 619, 601, 8821, 86364397717734821, 421950627598601, 2621, 295831, 47, 2237, 157, 53, 307, 7741, 6823, 61, 205883, 487, 67, 21767149, 71, 73, 149, 2004383, 79, 34033

Offset: 1

Zhi-Wei Sun, May 09 2014

nonn

Conjecture: (i) a(n) is prime for any n > 1.

(ii) For any prime p > 5, there exists a prime q < p/2 such that H(2,q-1) = sum_{0

See also A242222 and A242223 for similar conjectures involving harmonic numbers H(n) = sum_{k=1..n}1/k (n > 0).

			a(4) = 41 since H(2,4) = 5*41/(2^4*3^2) but none of H(2,1) = 1, H(2,2) = 5/2^2 and H(2,3) = 7^2/(2^2*3^2) is congruent to 0 modulo 41.

Zhi-Wei Sun, Table of n, a(n) for n = 1..108
Zhi-Wei Sun, Notes on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A007406, A007407, A242210, A242213, A242222, A242223.

h[n_]:=Numerator[HarmonicNumber[n,2]]
f[n_]:=FactorInteger[h[n]]
p[n_]:=p[n]=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[h[n]<2, Goto[cc]]; Do[Do[If[Mod[h[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n,1,40}]

Showing 1-3 of 3 results.

cp's OEIS Frontend

A242210 Number of primes p < prime(n) such that the Bernoulli number B_{p-1} is a primitive root modulo prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A242222 Number of primes p <= (prime(n)+1)/2 such that the harmonic number H(p-1) = sum_{0

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A242241 Least prime p such that H(2,n) = sum_{k=1..n}1/k^2 == 0 (mod p) but there is no 0 < k < n with H(2,k) == 0 (mod p), or 1 if such a prime p does not exist.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica