OFFSET
1,1
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
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.
FORMULA
A212793(a(n) - 1) = 1. - Reinhard Zumkeller, May 27 2012
EXAMPLE
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.
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 *)
PROG
(Haskell)
a097375 n = a097375_list !! (n-1)
a097375_list = filter ((== 1) . a212793 . (subtract 1)) a000040_list
-- Reinhard Zumkeller, May 27 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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, Dec 08 2003 and Reinhard Zumkeller, Aug 11 2004
EXTENSIONS
Corrected and extended by Harvey P. Dale, Feb 05 2012
STATUS
approved