A089199 Primes p such that p+1 is divisible by a cube.

7, 23, 31, 47, 53, 71, 79, 103, 107, 127, 151, 167, 191, 199, 223, 239, 263, 269, 271, 311, 359, 367, 383, 431, 439, 463, 479, 487, 499, 503, 593, 599, 607, 631, 647, 701, 719, 727, 743, 751, 809, 823, 839, 863, 887, 911, 919, 967, 971, 983, 991

Offset: 1

Author

Cino Hilliard, Dec 08 2003

Keywords

• easy
• nonn

Comments

This sequence is infinite and its relative density in the sequence of primes is equal to 1 - Product_{p prime} (1-1/(p^2*(p-1))) = 1 - A065414 = 0.302498... (Mirsky, 1949). - Amiram Eldar, Apr 07 2021

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

Includes A007522 and A141965.

Cf. A049098, A065414.

Programs

Maple

filter:= proc(p)
  isprime(p) and ormap(t -> t[2]>=3, ifactors(p+1)[2])
end proc:
select(filter, [seq(i,i=3..2000,2)]); # Robert Israel, Jan 11 2019

Mathematica

f[n_]:=Max[Last/@FactorInteger[n]]; lst={};Do[p=Prime[n];If[f[p+1]>=3,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)

PARI

ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) }
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) }