A089199 - OEIS

%I #25 Aug 26 2024 11:43:29

%S 7,23,31,47,53,71,79,103,107,127,151,167,191,199,223,239,263,269,271,

%T 311,359,367,383,431,439,463,479,487,499,503,593,599,607,631,647,701,

%U 719,727,743,751,809,823,839,863,887,911,919,967,971,983,991

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

%C 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

%H Robert Israel, <a href="/A089199/b089199.txt">Table of n, a(n) for n = 1..10000</a>

%H Leon Mirsky, <a href="https://www.jstor.org/stable/2305811">The number of representations of an integer as the sum of a prime and a k-free integer</a>, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 17-19.

%p filter:= proc(p)

%p   isprime(p) and ormap(t -> t[2]>=3, ifactors(p+1)[2])

%p end proc:

%p select(filter, [seq(i,i=3..2000,2)]); # _Robert Israel_, Jan 11 2019

%t 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 *)

%o (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) }

%o 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) }

%Y Includes A007522 and A141965.

%Y Cf. A049098, A065414.

%K easy,nonn

%O 1,1

%A _Cino Hilliard_, Dec 08 2003