A252232 a(n) = smallest prime q where exactly n primes p exist such that p < q and q^(p-1) == 1 (mod p^2), i.e., smallest prime base q having exactly n Wieferich primes less than q.

5, 17, 19, 233, 293, 977, 1451, 1693, 33301, 308093

Offset: 1

Author

Felix Fröhlich, Dec 15 2014

Keywords

• nonn
• hard
• more

Comments

From Robert G. Wilson v, Mar 11 2015: (Start)

n b p

1: 5 {2}

2: 17 {2, 3}

3: 19 {3, 7, 13}

4: 233 {2, 3, 11, 157}

5: 293 {2, 5, 7, 19, 83}

6: 977 {2, 11, 17, 109, 239, 401}

7: 1451 {5, 7, 11, 13, 83, 173, 1259}

8: 1693 {2, 3, 5, 11, 31, 37, 61, 109}

9: 33301 {2, 3, 5, 7, 43, 293, 317, 383, 1627}

10: 308093 {2, 5, 7, 11, 31, 47, 89, 167, 523, 619}

... (End)

Examples

			a(5) = 293, because q = 293 is the smallest prime for which there are exactly five primes p with p < q such that q^(p-1) == 1 (mod p^2), namely 2, 5, 7, 19 and 83.
		

Crossrefs

Cf. A175932, A222206.

For the values of p, see A252582.

Programs

PARI

for(n=1, 10, q=2; while(q > 1, q=nextprime(q+1); i=0; forprime(p=2, q, if(Mod(q, p^2)^(p-1)==1, i++); if(i==n, print1(q, ", "); break({2})))))