A089189 - OEIS

%I #35 Feb 16 2021 04:32:30

%S 2,3,5,7,11,13,19,23,29,31,37,43,47,53,59,61,67,71,79,83,101,103,107,

%T 127,131,139,149,151,157,167,173,179,181,191,197,199,211,223,227,229,

%U 239,263,269,277,283,293,307,311,317,331,347,349,359,367,373,383

%N Primes p such that p-1 is cubefree.

%C 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.

%C 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

%H Reinhard Zumkeller and Vincenzo Librandi, <a href="/A089189/b089189.txt">Table of n, a(n) for n = 1..10000</a>

%H Leon Mirsky, <a href="http://www.jstor.org/stable/2305811">The number of representations of an integer as the sum of a prime and a k-free integer</a>, American Mathematial Monthly 56:1 (1949), pp. 17-19.

%F A212793(a(n) - 1) = 1. - _Reinhard Zumkeller_, May 27 2012

%e 43 is included because 43-1 = 2*3*7.

%e 41 is omitted because 41-1 = 2^3*5.

%e 97 is omitted because 96 = 2^5*3 since higher powers are also tested for exclusion.

%p filter:= p -> isprime(p) and max(seq(t[2],t=ifactors(p-1)[2]))<=2:

%p select(filter, [2,seq(2*i+1,i=1..1000)]); # _Robert Israel_, Sep 11 2014

%t 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 *)

%t Select[Prime[Range[100]],Max[Transpose[FactorInteger[#-1]][[2]]]<3&] (* _Harvey P. Dale_, Feb 05 2012 *)

%o (Haskell)

%o a097375 n = a097375_list !! (n-1)

%o a097375_list = filter ((== 1) . a212793 . (subtract 1)) a000040_list

%o -- _Reinhard Zumkeller_, May 27 2012

%o (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

%Y Cf. A004709, A039787, A065414, A097380, A089194 (subsequence).

%K easy,nonn

%O 1,1

%A _Cino Hilliard_, Dec 08 2003 and _Reinhard Zumkeller_, Aug 11 2004

%E Corrected and extended by _Harvey P. Dale_, Feb 05 2012