A049097 Primes p such that p+1 is squarefree.

2, 5, 13, 29, 37, 41, 61, 73, 101, 109, 113, 137, 157, 173, 181, 193, 229, 257, 277, 281, 313, 317, 353, 373, 389, 397, 401, 409, 421, 433, 457, 461, 509, 541, 569, 601, 613, 617, 641, 653, 661, 673, 677, 709, 733, 757, 761, 769, 797, 821, 829, 853, 857

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Numbers k such that core(sigma(k)) = k + 1 where core(k) is the squarefree part of k (A007913). - Benoit Cloitre, May 01 2002

This sequence is infinite and its relative density in the sequence of primes is equal to Artin's constant (A005596): Product_{p prime} (1-1/(p*(p-1))) = 0.373955... (Mirsky, 1949). - Amiram Eldar, Dec 29 2020

Examples

			29 is included since 29 + 1 = 30 = 2*3*5 is squarefree.
17 is not here because 18 is divisible by a square, 9.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 17-19.

Crossrefs

Cf. A000040, A005117, A039787.

Cf. A000203, A005596, A007913, A077067, A160696.

Programs

Magma

[ p: p in PrimesUpTo(900) | IsSquarefree(p+1) ]; // Vincenzo Librandi, Dec 25 2010

Maple

N:= 10000; # to get all entries up to N
A049097:= select(t -> isprime(t) and numtheory:-issqrfree(t+1), [2, seq(1+2*k,k=1..floor((N-1)/2))]); # Robert Israel, May 11 2014

Mathematica

Select[Prime[Range[100]], SquareFreeQ[# + 1] &] (* Zak Seidov, Feb 08 2016 *)

PARI

lista(nn) = forprime(p=1, nn, if (issquarefree(p+1), print1(p, ", "))); \\ Michel Marcus, Jan 08 2015

Formula

A160696(a(n)) = 1. - Reinhard Zumkeller, May 24 2009

a(n) = A077067(n)-1. - Zak Seidov, Mar 19 2016