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.

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

Author

Zhi-Wei Sun, May 22 2014

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Mathematica

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}]