A215726 -id:A215726 - cp's OEIS frontend

A061304 Squarefree triangular numbers.

1, 3, 6, 10, 15, 21, 55, 66, 78, 91, 105, 190, 210, 231, 253, 406, 435, 465, 561, 595, 703, 741, 861, 903, 946, 1081, 1326, 1378, 1653, 1711, 1770, 1830, 1891, 2145, 2211, 2278, 2346, 2415, 2485, 2701, 2926, 3003, 3081, 3403, 3486, 3570, 3655, 3741, 4186, 4278

Offset: 1

Amarnath Murthy, Apr 26 2001

			105 = 3 * 5 * 7 is a squarefree triangular number.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000217, A215726.

# uses code of A000217
isA061304 := proc(n)
    isA000217(n) and issqrfree(n) ;
    simplify(%) ;
end proc:
for n from 1 to 5000 do
    if isA061304(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Oct 05 2017

Select[Accumulate[Range[0, 100]], SquareFreeQ] (* Jean-François Alcover, Apr 17 2020 *)

isA078779F(f)=for(i=2,#f~, if(f[i,2]>1, return(0))); #f~==0 || f[1,2]==1 || (f[1,2]==2 && f[1,1]==2)
list(lim)=my(v=List(), ok=1); forfactored(n=2, (sqrtint(lim\1*8+1)+1)\2, e=n[2][,2]; if(isA078779F(n[2]), if(ok, listput(v, binomial(n[1],2)), ok=1), ok=0)); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

A010054(a(n))*A008966(a(n)) = 1. - Reinhard Zumkeller, Nov 01 2009

a(n) = A000217(A215726(n)). - Zak Seidov, Aug 22 2012

A067197 Numbers k such that k*(k+1)/2 is not squarefree.

Original entry on oeis.org

7, 8, 9, 15, 16, 17, 18, 23, 24, 25, 26, 27, 31, 32, 35, 36, 39, 40, 44, 45, 47, 48, 49, 50, 53, 54, 55, 56, 62, 63, 64, 71, 72, 74, 75, 79, 80, 81, 87, 88, 89, 90, 95, 96, 97, 98, 99, 100, 103, 104, 107, 108, 111, 112, 116, 117, 119, 120, 121, 124, 125, 126, 127, 128

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

The asymptotic density of this sequence is 1 - (3/2)*A065474 = 0.5160488515... (Granville and Ramaré, 1996). - Amiram Eldar, Mar 02 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
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.

Complement of A215726.

Cf. A000217, A005117, A013929, A065474.

Select[Range[200],!SquareFreeQ[#(#+1)/2]&] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)

isok(k) = !issquarefree(k*(k+1)/2); \\ Michel Marcus, Feb 17 2021

A175608 Characteristic function of squarefree triangular integers: 1 if n(n+1)/2 is squarefree else 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1

Offset: 1

Zak Seidov, Jul 23 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008966 (1 if n is squarefree, else 0), A010051 (Characteristic function of primes: 1 if n is prime else 0).

Cf. A000217, A061304, A065474, A215726.

Table[If[SquareFreeQ[n(n+1)/2],1,0],{n,200}]
If[SquareFreeQ[#],1,0]&/@Accumulate[Range[120]] (* Harvey P. Dale, Nov 22 2022 *)

a(n) = A008966(n(n+1)/2).

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = (3/2)*A065474 = 0.4839511484... . - Amiram Eldar, May 10 2022

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A061304 Squarefree triangular numbers.

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A067197 Numbers k such that k*(k+1)/2 is not squarefree.

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A175608 Characteristic function of squarefree triangular integers: 1 if n(n+1)/2 is squarefree else 0.

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