A089191 Primes p such that p+1 is cubefree.

2, 3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 59, 61, 67, 73, 83, 89, 97, 101, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 233, 241, 251, 257, 277, 281, 283, 293, 307, 313, 317, 331, 337, 347, 349, 353, 373, 379, 389, 397, 401, 409

Offset: 1

Author

Cino Hilliard, Dec 08 2003

Keywords

• easy
• nonn

Comments

The ratio of the count of primes p <= n such that p+1 is cubefree to the count of primes <= n converges to 0.69+ slightly higher than the p-1 variety.

More accurately, the density of this sequence within the primes is Product_{p prime} (1-1/(p^2*(p-1))) = 0.697501... (A065414) (Mirsky, 1949). - Amiram Eldar, Feb 16 2021

Examples

			43 is included because 43+1 = 2^2*11.
71 is omitted because 71+1 = 2^3*3^2.
		

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, American Mathematial Monthly 56:1 (1949), pp. 17-19.

Crossrefs

Cf. A004709, A049097, A065414, A089189.

Programs

Maple

filter:= t -> isprime(t) and max(map(s -> s[2], ifactors(t+1)[2]))<3:
select(filter, [2,seq(i,i=3..1000,2)]); # Robert Israel, Mar 18 2018

Mathematica

Select[Prime[Range[100]],Max[Transpose[FactorInteger[#+1]][[2]]]<3&] (* Harvey P. Dale, Jun 06 2013 *)

PARI

is(n) = isprime(n) && vecmax(factor(n+1)[,2]) < 3 \\ Amiram Eldar, Feb 16 2021