A069977 Numbers k such that k and k+2 are squarefree.

1, 3, 5, 11, 13, 15, 17, 19, 21, 29, 31, 33, 35, 37, 39, 41, 51, 53, 55, 57, 59, 65, 67, 69, 71, 77, 83, 85, 87, 89, 91, 93, 95, 101, 103, 105, 107, 109, 111, 113, 127, 129, 131, 137, 139, 141, 143, 149, 155, 157, 159, 161, 163, 165, 177, 179, 181, 183, 185, 191, 193

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 01 2002

Keywords

• nonn

Comments

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474) (Mirsky, 1949). - Amiram Eldar, Dec 29 2020

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
• Bin Chen, On almost-prime $k$-tuples, arXiv preprint (2023). arXiv:2301.05044 [math.NT], 2023.
• Leon Mirsky, On the frequency of pairs of square-free numbers with a given difference, Bull. Amer. Math. Soc., Vo. 55, No. 10 (1949), pp. 936-939.

Crossrefs

Cf. A007674, A065474.

Programs

Mathematica

f[n_] := Union[ Transpose[ FactorInteger[n]] [[ -1]]] [[ -1]]; Select[ Range[200], f[ # ] < 2 && f[ # + 2] < 2 & ]
SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,200}],{1,,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Oct 06 2017 *)

PARI

is(n)=n%2 && issquarefree(n) && issquarefree(n+2) \\ Charles R Greathouse IV, May 15 2016

Extensions

Edited and extended by Robert G. Wilson v, May 04 2002

1 prepended by Vladimir Joseph Stephan Orlovsky, Mar 30 2011