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

1,2

Comments

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

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.

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

A007674 is a subsequence.

Cf. A000217, A005117, A061304, A065474.

Sequence in context: A047423 A032970 A137691 * A158520 A032868 A032341

Adjacent sequences: A215723 A215724 A215725 * A215727 A215728 A215729

Keyword

nonn

Author

Zak Seidov, Aug 22 2012

Status

approved