A049092 Primes p such that p-1 is not squarefree.

5, 13, 17, 19, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 127, 137, 149, 151, 157, 163, 173, 181, 193, 197, 199, 229, 233, 241, 251, 257, 269, 271, 277, 281, 293, 307, 313, 317, 337, 349, 353, 373, 379, 389, 397, 401, 409, 421, 433, 449, 457, 461, 487

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Primes p with mu(p-1)=0, where mu is the Möbius function. - T. D. Noe, Nov 03 2003

Primes p such that the sum of the primitive roots of p (see A088144) is 0 mod p. - Jon Wharf, Mar 12 2015

The relative density of this sequence within the primes is 1 - A005596 = 0.626044... - Amiram Eldar, Feb 10 2021

Examples

			p = 257 is here because p-1 = 256 = 2^8.
p = 997 is here because p-1 = 996 = 3*(2^2)*83.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Eric Weisstein's World of Mathematics, Moebius Function.

Crossrefs

Cf. A005596, A039787, A078330 (primes p with mu(p-1)=-1), A088179 (primes p such that mu(p-1)=1), A089451 (mu(p-1) for prime p), A145199.

Programs

Magma

[ p: p in PrimesUpTo(500) | not IsSquarefree(p-1) ]; // Vincenzo Librandi, Mar 12 2015

Mathematica

Select[Prime[Range[400]], MoebiusMu[ #-1]==0&]

PARI

forprime(p=2,500,if(!issquarefree(p-1),print(p))) \\ Michael B. Porter, Mar 16 2015

Formula

a(n) = A145199(n) + 1. - Amiram Eldar, Feb 10 2021