A215726 Numbers k such that the k-th triangular number is squarefree.

1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 19, 20, 21, 22, 28, 29, 30, 33, 34, 37, 38, 41, 42, 43, 46, 51, 52, 57, 58, 59, 60, 61, 65, 66, 67, 68, 69, 70, 73, 76, 77, 78, 82, 83, 84, 85, 86, 91, 92, 93, 94, 101, 102, 105, 106, 109, 110, 113, 114, 115, 118, 122, 123

Offset: 1

Author

Zak Seidov, Aug 22 2012

Keywords

• nonn

Comments

The asymptotic density of this sequence is (3/2)*A065474 = 0.4839511484... (Granville and Ramaré, 1996). - Amiram Eldar, Feb 17 2021

Examples

			14 is a term because A000217(14) = 14*15/2 = 105 = 3*5*7.
		

References

• Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 184.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)
• Andrew Granville and Olivier Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika, Vol. 43, No. 1 (1996), pp. 73-107; alternative link.

Crossrefs

A007674 is a subsequence.

Cf. A000217, A005117, A061304, A065474.

Programs

Mathematica

Select[Range[123],SquareFreeQ[#(#+1)/2]&]
Position[Accumulate[Range[150]],?(SquareFreeQ[#]&)]//Flatten//Rest (* _Harvey P. Dale, Jul 07 2020 *)

PARI

is(n)=issquarefree(n/gcd(n,2))&&issquarefree((n+1)/gcd(n+1,2)) \\ Charles R Greathouse IV, Jun 06 2013

Formula

Numbers k such that A000217(k) is squarefree. [corrected by Zak Seidov, Jun 05 2013]