A089189 Primes p such that p-1 is cubefree.

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 127, 131, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 239, 263, 269, 277, 283, 293, 307, 311, 317, 331, 347, 349, 359, 367, 373, 383

Offset: 1

Author

Cino Hilliard, Dec 08 2003 and Reinhard Zumkeller, Aug 11 2004

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.. . This implies that roughly 70% of the primes less one are cubefree. This compares to about 0.37 of the primes less one are squarefree.

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*3*7.
41 is omitted because 41-1 = 2^3*5.
97 is omitted because 96 = 2^5*3 since higher powers are also tested for exclusion.
		

Links

• Reinhard Zumkeller and Vincenzo Librandi, 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, A039787, A065414, A097380, A089194 (subsequence).

Programs

Haskell

a097375 n = a097375_list !! (n-1)
a097375_list = filter ((== 1) . a212793 . (subtract 1)) a000040_list
-- Reinhard Zumkeller, May 27 2012

Maple

filter:= p -> isprime(p) and max(seq(t[2],t=ifactors(p-1)[2]))<=2:
select(filter, [2,seq(2*i+1,i=1..1000)]); # Robert Israel, Sep 11 2014

Mathematica

f[n_]:=Module[{a=m=0},Do[If[FactorInteger[n][[m,2]]>2,a=1],{m,Length[FactorInteger[n]]}];a]; lst={};Do[p=Prime[n];If[f[p-1]==0,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 15 2009 *)
Select[Prime[Range[100]],Max[Transpose[FactorInteger[#-1]][[2]]]<3&] (* Harvey P. Dale, Feb 05 2012 *)

PARI

lista(nn) = forprime(p=2, nn, f = factor(p-1)[,2]; if ((#f == 0) || vecmax(f) < 3, print1(p, ", "));) \\ Michel Marcus, Sep 11 2014

Formula

A212793(a(n) - 1) = 1. - Reinhard Zumkeller, May 27 2012

Extensions

Corrected and extended by Harvey P. Dale, Feb 05 2012