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

1,2

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.

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 *)

Prog

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

Crossrefs

Cf. A007674, A065474.

Sequence in context: A014597 A357369 A130603 * A284795 A329629 A340019

Adjacent sequences: A069974 A069975 A069976 * A069978 A069979 A069980

Keyword

nonn

Author

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

Extensions

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

1 prepended by Vladimir Joseph Stephan Orlovsky, Mar 30 2011

Status

approved