A242748 -id:A242748 - cp's OEIS frontend

A242750 Positive integers n with property that n is a primitive root modulo prime(n).

1, 2, 3, 6, 7, 10, 11, 13, 15, 18, 24, 26, 28, 33, 39, 41, 44, 45, 48, 50, 54, 55, 56, 58, 62, 65, 68, 69, 71, 75, 79, 83, 85, 93, 95, 107, 108, 109, 110, 117, 118, 119, 120, 123, 126, 129, 130, 131, 133, 139, 142, 143, 145, 148, 157, 158, 160, 163, 164, 166, 170, 172, 173, 174, 179, 182, 186, 190, 191, 195

Offset: 1

Zhi-Wei Sun, May 21 2014

nonn

According to the conjecture in A242748, this sequence should have infinitely many terms.

			6 is a member since 6 is a primitive root modulo prime(6) = 13, but 4 and 5 are not since 4 is not a primitive root modulo prime(4) = 7 and 5 is not a primitive root modulo prime(5) = 11.

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

Cf. A000040, A000720, A242748, A242752.

dv[n_]:=Divisors[n]
n=0;Do[Do[If[Mod[k^(Part[dv[Prime[k]-1],j]),Prime[k]]==1,Goto[aa]],{j,1,Length[dv[Prime[k]-1]]-1}];n=n+1;Print[n," ",k];Label[aa];Continue,{k,1,195}]

A242752 Primes p such that pi(p) is a primitive root modulo p, where pi(p) is the number of primes not exceeding p.

Original entry on oeis.org

2, 3, 5, 13, 17, 29, 31, 41, 47, 61, 89, 101, 107, 137, 167, 179, 193, 197, 223, 229, 251, 257, 263, 271, 293, 313, 337, 347, 353, 379, 401, 431, 439, 487, 499, 587, 593, 599, 601, 643, 647, 653, 659, 677, 701, 727, 733, 739, 751, 797, 821, 823, 829, 857, 919, 929, 941, 967, 971, 983

Offset: 1

Zhi-Wei Sun, May 21 2014

nonn

According to the conjecture in A232748, this sequence should contain infinitely many primes.

			a(3) = 5 since 5 is prime with pi(5) = 3 a primitive root modulo 5.

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

Cf. A000040, A000720, A242748, A242750.

dv[n_]:=Divisors[n]
n=0;Do[Do[If[Mod[k^(Part[dv[Prime[k]-1],j]),Prime[k]]==1,Goto[aa]],{j,1,Length[dv[Prime[k]-1]]-1}];n=n+1;Print[n," ",Prime[k]];Label[aa];Continue,{k,1,166}]

A242753 Number of ordered ways to write n = k + m with 0 < k <= m such that the inverse of k mod prime(k) among 1, ..., prime(k) - 1 is prime and the inverse of m mod prime(m) among 1, ..., prime(m) - 1 is also prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 3, 1, 2, 4, 3, 2, 3, 4, 2, 1, 2, 3, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 2, 1, 2, 3, 3, 4, 1, 1, 3, 4, 2, 4, 4, 5, 3, 4, 5, 4, 3, 5, 6, 3, 3, 6, 4, 4, 3, 5, 4, 4, 4, 6, 5, 3, 5, 6, 5, 5, 9, 5, 6, 4

Offset: 1

Zhi-Wei Sun, May 22 2014

nonn

Conjecture: a(n) > 0 for all n > 3.

This implies that there are infinitely many positive integers k such that k*q == 1 (mod prime(k)) for some prime q < prime(k).

			a(11) = 1 since 11 = 4 + 7, 4*2 == 1 (mod prime(4)=7) with 2 prime, and 7*5 == 1 (mod Prime(7)=17) with 5 prime.
a(36) = 1 since 36 = 18 + 18, and 18*17 == 1 (mod 61) with 17 prime.
a(46) = 1 since 46 = 6 + 40, 6*11 == 1 (mod prime(6)= 13) with 11 prime, and 40*13 == 1 (mod prime(40)=173) with 13 prime.

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

Cf. A000040, A000720, A242425, A242748, A242750, A242752, A242754, A242755.

p[n_]:=PrimeQ[PowerMod[n,-1,Prime[n]]]
Do[m=0;Do[If[p[k]&&p[n-k],m=m+1],{k,1,n/2}];Print[n," ",m];Continue,{n,1,80}]

A243164 Number of primes p < n such that p*n is a primitive root modulo prime(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 2, 3, 4, 2, 2, 4, 3, 3, 5, 3, 5, 5, 6, 2, 5, 5, 4, 4, 2, 4, 6, 7, 6, 7, 2, 7, 4, 7, 4, 4, 6, 7, 3, 7, 7, 3, 7, 7, 9, 7, 6, 5, 6, 6, 7, 7, 9, 4, 9, 8, 2, 10, 7, 9, 11, 5, 6, 5, 9, 11, 8, 6, 9

Offset: 1

Zhi-Wei Sun, May 31 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 6.
(ii) Any integer n > 6 can be written as k + m with k > 0 and m > 0 such that k*m is a primitive root modulo prime(n).
We have verified part (i) for all n = 7, ..., 2*10^5.

			a(4) = 1 since 3 is prime with 3*4 = 12 a primitive root modulo prime(4) = 7.
a(9) = 1 since 7 is prime with 7*9 = 63 a primitive root modulo prime(9) = 23.
a(10) = 1 since 5 is prime with 5*10 = 50 a primitive root modulo prime(10) = 29.
a(12) = 1 since 2 is prime with 2*12 = 24 a primitive root modulo prime(12) = 37.

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

Cf. A000040, A000720, A237497, A237578, A242748.

dv[n_]:=Divisors[n]
Do[m=0;Do[Do[If[Mod[(Prime[k]*n)^(Part[dv[Prime[n]-1],i]),Prime[n]]==1,Goto[aa]],{i,1,Length[dv[Prime[n]-1]]-1}];m=m+1;Label[aa];Continue,{k,1,PrimePi[n-1]}];Print[n," ",m];Continue,{n,1,80}]

A242754 Positive integers k such that k*p == 1 (mod prime(k)) for some prime p < prime(k).

Original entry on oeis.org

2, 3, 4, 6, 7, 10, 11, 13, 17, 18, 21, 31, 37, 40, 41, 46, 48, 49, 52, 53, 58, 60, 64, 66, 70, 71, 72, 73, 75, 81, 85, 92, 93, 96, 100, 102, 109, 117, 119, 127, 136, 137, 140, 143, 145, 146, 149, 160, 162, 179, 189, 194, 200, 206, 215, 232, 233, 243, 246, 247

Offset: 1

Zhi-Wei Sun, May 22 2014

nonn

According to the conjecture in A242753, this sequence should have infinitely many terms.

Conjecture: The number of terms not exceeding x > 1 has the main term x/(log x) as x tends to infinity.

			a(4) = 6 since 6*11 == 1 (mod prime(6)=13) with 11 prime, but 5*9 == 1 (mod prime(5)=11) with 9 composite.

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

Cf. A000040, A000720, A242425, A242748, A242750, A242752, A242753, A242755.

p[n_]:=PrimeQ[PowerMod[n,-1,Prime[n]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",k]];Continue,{k,1,247}]

A242755 Primes p such that pi(p)*q == 1 (mod p) for some prime q < p, where pi(p) is the number of primes not exceeding p.

Original entry on oeis.org

3, 5, 7, 13, 17, 29, 31, 41, 59, 61, 73, 127, 157, 173, 179, 199, 223, 227, 239, 241, 271, 281, 311, 317, 349, 353, 359, 367, 379, 419, 439, 479, 487, 503, 541, 557, 599, 643, 653, 709, 769, 773, 809, 823, 829, 839, 859, 941, 953, 1063

Offset: 1

Zhi-Wei Sun, May 22 2014

nonn

According to the conjecture in A242753, this sequence should contain infinitely many primes.

Conjecture: The number of such primes not exceeding x > 1 has the main term x/(log x)^2 as x tends to infinity.

			a(4) = 13 since 13 is prime with pi(13) = 6, and 6*11 == 1 (mod 13) with 11 prime, but pi(11)*9 == 1 (mod 11) with 9 not prime.

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

Cf. A000040, A000720, A242748, A242750, A242752, A242753, A242754.

p[n_]:=PrimeQ[PowerMod[n,-1,Prime[n]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]];Continue,{k,1,179}]

A242950 Number of ordered ways to write n = k + m with k > 1 and m > 1 such that the least nonnegative residue of prime(k) modulo k is a square and the least nonnegative residue of prime(m) modulo m is a prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 1, 3, 2, 1, 1, 3, 4, 4, 1, 3, 5, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 3, 5, 2, 5, 3, 5, 3, 6, 3, 7, 4, 6, 5, 7, 5, 9, 7, 6, 4, 6, 5, 9, 5, 6, 8, 7, 8, 5, 8, 5, 8, 4, 8, 6, 7, 4, 7, 4, 6, 4, 5, 4, 8, 2, 3, 4, 5, 4, 5, 6, 7, 7

Offset: 1

Zhi-Wei Sun, May 27 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 7.
(ii) Any integer n > 9 can be written as k + m with k > 1 and m > 1 such that the least nonnegative residue of prime(k) modulo k and the least nonnegative residue of prime(m) modulo m are both prime.
We have verified a(n) > 0 for all n = 8, ..., 10^8.

			a(11) = 1 since 11 = 2 + 9, prime(2) = 3 == 1^2 (mod 2), and prime(9) = 23 == 5 (mod 9) with 5 prime.
a(16) = 1 since 16 = 12 + 4, prime(12) = 37 == 1^2 (mod 12), and prime(4) = 7 == 3 (mod 4) with 3 prime.

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

Cf. A000290, A000040, A242425, A242748, A242753.

SQ[n_]:=IntegerQ[Sqrt[n]]
s[k_]:=SQ[Mod[Prime[k],k]]
p[k_]:=PrimeQ[Mod[Prime[k],k]]
a[n_]:=Sum[Boole[s[k]&&p[n-k]],{k,2,n-2}]
Table[a[n],{n,1,80}]

A242879 Least positive integer k < n such that kp == 1 (mod prime(k)) for some prime p < prime(k) and (n-k)q == 1 (mod prime(n-k)) for some prime q < prime(n-k), or 0 if such a number k does not exist.

Original entry on oeis.org

0, 0, 0, 2, 2, 2, 3, 2, 2, 3, 4, 2, 2, 3, 2, 3, 4, 7, 2, 2, 3, 4, 2, 3, 4, 13, 6, 7, 11, 13, 10, 11, 2, 3, 4, 18, 6, 7, 2, 3, 4, 2, 2, 3, 4, 6, 6, 2, 3, 2, 2, 3, 4, 2, 2, 3, 4, 6, 6, 2, 3, 2, 3, 4, 7, 2, 3, 2, 3, 4, 7, 2, 2, 2, 2, 3, 2, 3, 4, 7

Offset: 1

Zhi-Wei Sun, May 25 2014

nonn

According to the conjecture in A242753, a(n) should be positive for all n > 3.

We have verified that a(n) > 0 for all n = 4, ..., 10^8.

			a(4) = 2 since 4 = 2 + 2 and 2*2 == 1 (mod prime(2)=3).
a(7) = 3 since 7 = 3 + 4, 3*2 == 1 (mod prime(3)=5) with 2 prime, and also 4*2 == 1 (mod prime(4)=7) with 2 prime, but 5*9 == 1 (mod prime(5)=11) with 9 not prime.

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

Cf. A000040, A242425, A242748, A242753, A242754, A242755.

p[n_]:=PrimeQ[PowerMod[n,-1,Prime[n]]]
Do[Do[If[p[k]&&p[n-k],Print[n," ",k];Goto[aa]];Continue,{k,1,n/2}];Print[n," ",0];Label[aa];Continue,{n,1,80}]

A261387 Number of ways to write n = k + m with 0 < k < m < n such that prime(k) is a primitive root modulo prime(m) and also prime(m) is a primitive root modulo prime(k).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 0, 2, 1, 3, 3, 1, 1, 2, 1, 2, 7, 4, 2, 1, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 7, 3, 3, 5, 5, 5, 5, 4, 3, 6, 7, 5, 5, 5, 3, 7, 7, 5, 2, 7, 6, 4, 5, 5, 7, 10, 9, 8, 8, 4, 7, 5, 11, 14, 7, 12, 11, 9, 6

Offset: 1

Zhi-Wei Sun, Aug 27 2015

nonn

Conjecture: (i) a(n) > 0 except for n = 1, 2, 8.

(ii) Any positive rational number r not equal to 1 can be written as m/n, where m and n are positive integers such that prime(m) is a primitive root modulo prime(n) and also prime(n) is a primitive root modulo prime(m).

			a(7) = 2 since 7 = 1+6 = 3+4, prime(1) = 2 is a primitive root modulo prime(6) = 13 and 13 is a primitive root modulo 2, also prime(3) = 5 is a primitive root modulo prime(4) = 7 and 7 is a primitive root modulo 5.
a(22) = 1 since 22 = 4+18, prime(4)= 7 is a primitive root modulo prime(18) = 61 and 61 is a primitive root modulo 7.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Zhi-Wei Sun, Checking part (ii) of the conjecture for r = a/b with 1 <= a < b <= 100
Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A242748, A259492.

f[n_]:=Prime[n]
Dv[n_]:=Divisors[n]
LL[n_]:=Length[Dv[n]]
Do[r=0;Do[Do[If[Mod[f[k]^(Part[Dv[f[n-k]-1],i])-1,f[n-k]]==0,Goto[bb]],{i,1,LL[f[n-k]-1]-1}];Do[If[Mod[f[n-k]^(Part[Dv[f[k]-1],i])-1,f[k]]==0,Goto[bb]],{i,1,LL[f[k]-1]-1}];
r=r+1;Label[bb];Continue,{k,1,(n-1)/2}];Print[n," ",r];Continue,{n,1,70}]

A293213 Primes p with phi(p-1) a primitive root modulo p, where phi(.) is Euler's totient function (A000010).

Original entry on oeis.org

2, 5, 23, 43, 47, 67, 101, 149, 167, 211, 229, 263, 269, 281, 349, 353, 359, 383, 389, 421, 431, 449, 461, 479, 499, 503, 509, 521, 661, 691, 709, 719, 739, 743, 829, 839, 859, 863, 883, 887, 907, 941, 953, 971, 983, 991, 1031, 1087, 1103, 1109, 1163, 1181, 1229, 1237, 1279, 1291, 1319, 1327, 1367, 1373

Offset: 1

Zhi-Wei Sun, Oct 02 2017

nonn

It is well known that for any prime p the number of distinct primitive roots modulo p among 1,...,p-1 is phi(p-1).

Conjecture: The sequence contains infinitely many terms. Moreover, the number of primes p <= x with phi(p-1) a primitive root modulo p is asymptotically equivalent to c*x/(log x) as x tends to the infinity, where c is a constant with 0.36 < c < 0.37.

			a(2) = 5 since phi(5-1) = 2 is a primitive root modulo the prime 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Cf. A000010, A000040, A242748, A242750, A291615, A291657.

p[n_]:=p[n]=Prime[n];
n=0;Do[Do[If[Mod[EulerPhi[p[k]-1]^(Part[Divisors[p[k]-1],i])-1,p[k]]==0,Goto[aa]],{i,1,Length[Divisors[p[k]-1]]-1}];
n=n+1;Print[n," ",p[k]];Label[aa],{k,1,220}]

cp's OEIS Frontend

A242750 Positive integers n with property that n is a primitive root modulo prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A242752 Primes p such that pi(p) is a primitive root modulo p, where pi(p) is the number of primes not exceeding p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A242753 Number of ordered ways to write n = k + m with 0 < k <= m such that the inverse of k mod prime(k) among 1, ..., prime(k) - 1 is prime and the inverse of m mod prime(m) among 1, ..., prime(m) - 1 is also prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A243164 Number of primes p < n such that p*n is a primitive root modulo prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A242754 Positive integers k such that k*p == 1 (mod prime(k)) for some prime p < prime(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A242755 Primes p such that pi(p)*q == 1 (mod p) for some prime q < p, where pi(p) is the number of primes not exceeding p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A242950 Number of ordered ways to write n = k + m with k > 1 and m > 1 such that the least nonnegative residue of prime(k) modulo k is a square and the least nonnegative residue of prime(m) modulo m is a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A242879 Least positive integer k < n such that k*p == 1 (mod prime(k)) for some prime p < prime(k) and (n-k)*q == 1 (mod prime(n-k)) for some prime q < prime(n-k), or 0 if such a number k does not exist.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A242879 Least positive integer k < n such that kp == 1 (mod prime(k)) for some prime p < prime(k) and (n-k)q == 1 (mod prime(n-k)) for some prime q < prime(n-k), or 0 if such a number k does not exist.