A049232 Primes p such that p+2 is divisible by a square.

2, 7, 23, 43, 47, 61, 73, 79, 97, 151, 167, 173, 223, 241, 277, 313, 331, 349, 359, 367, 373, 421, 439, 457, 523, 547, 601, 619, 673, 691, 709, 727, 733, 773, 823, 839, 853, 907, 929, 997, 1033, 1051, 1069, 1087, 1123, 1181, 1213, 1223, 1231, 1249, 1303

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

This sequence is infinite and its relative density in the sequence of the primes is equal to 1 - 2 * Product_{p prime} (1-1/(p*(p-1))) = 1 - 2 * A005596 = 0.252088... - Amiram Eldar, Feb 27 2021

Examples

			47 is a term since 47+2 = 49 = 7^2 is a square.
523 is a term since 523+2 = 525 = 5^2*21 is divisible by a square.
		

Crossrefs

A091880 gives prime indices.

Cf. A005596, A013929, A049229, A049233.

Programs

Mathematica

Select[Prime[Range[100]], ! SquareFreeQ[ # + 2] &] (* Zak Seidov, Oct 28 2008 *)

PARI

powerfreep3(n,p,k) = { c=0; pc=0; forprime(x=2,n, pc++; if(ispowerfree(x+k,p)==0, c++; print1(x","); ) ); print(); print(c","pc","c/pc+.0) }
ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) }

Formula

Primes p such that mu(p+2) = 0.

Extensions

Corrected by Cino Hilliard and Ray Chandler, Dec 08 2003

Edited by N. J. A. Sloane, Oct 27 2008 at the suggestion of R. J. Mathar