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

%I A089199 #25 Aug 26 2024 11:43:29
%S A089199 7,23,31,47,53,71,79,103,107,127,151,167,191,199,223,239,263,269,271,
%T A089199 311,359,367,383,431,439,463,479,487,499,503,593,599,607,631,647,701,
%U A089199 719,727,743,751,809,823,839,863,887,911,919,967,971,983,991
%N A089199 Primes p such that p+1 is divisible by a cube.
%C A089199 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 A089199 Robert Israel, <a href="/A089199/b089199.txt">Table of n, a(n) for n = 1..10000</a>
%H A089199 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 A089199 filter:= proc(p)
%p A089199   isprime(p) and ormap(t -> t[2]>=3, ifactors(p+1)[2])
%p A089199 end proc:
%p A089199 select(filter, [seq(i,i=3..2000,2)]); # _Robert Israel_, Jan 11 2019
%t A089199 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 A089199 (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 A089199 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 A089199 Includes A007522 and A141965.
%Y A089199 Cf. A049098, A065414.
%K A089199 easy,nonn
%O A089199 1,1
%A A089199 _Cino Hilliard_, Dec 08 2003