

A089189


Primes p such that p1 is cubefree.


11



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
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OFFSET

1,1


COMMENTS

The ratio of the count of primes p <= n such that p1 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} (11/(p^2*(p1))) = 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 kfree integer, American Mathematial Monthly 56:1 (1949), pp. 1719.


FORMULA

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


EXAMPLE

43 is included because 431 = 2*3*7.
41 is omitted because 411 = 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(p1)[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[p1]==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 !! (n1)
a097375_list = filter ((== 1) . a212793 . (subtract 1)) a000040_list
 Reinhard Zumkeller, May 27 2012
(PARI) lista(nn) = forprime(p=2, nn, f = factor(p1)[, 2]; if ((#f == 0)  vecmax(f) < 3, print1(p, ", ")); ) \\ Michel Marcus, Sep 11 2014


CROSSREFS

Cf. A004709, A039787, A065414, A097380, A089194 (subsequence).
Sequence in context: A057447 A095074 A042987 * A097375 A007459 A129944
Adjacent sequences: A089186 A089187 A089188 * A089190 A089191 A089192


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



