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A215726 Numbers k such that the k-th triangular number is squarefree. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 27 16:48 EDT 2022. Contains 354110 sequences. (Running on oeis4.)