A065474 -id:A065474 - cp's OEIS frontend

A117204 Squarefree positive integers k such that 2*k+1 is also squarefree.

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 23, 26, 29, 30, 33, 34, 35, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 59, 61, 65, 66, 69, 70, 71, 74, 77, 78, 79, 82, 83, 86, 89, 91, 93, 95, 97, 101, 102, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119

Offset: 1

Leroy Quet, Mar 02 2006

nonn

The asymptotic density of this sequence is (3/2)*A065474 = 0.4839511484... (Erdős and Ivić, 1987). - Amiram Eldar, Mar 02 2021

			10 and 2*10 +1 = 21 are both squarefree, so 10 is in the sequence.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Colloq. Math. Soc. János Bolyai, Vol. 51 (1987), pp. 45-91.

Cf. A005117, A117203, A117206.

Cf. A005384, A111153, A065474.

sfQ[n_]:=SquareFreeQ[n]&&SquareFreeQ[2n+1]; Select [Range[200],sfQ] (* Harvey P. Dale, Mar 12 2011 *)

a(n) = (A117203(n) - 1)/2.

More terms from Jonathan Vos Post, Mar 03 2006

Corrected and extended by Harvey P. Dale, Mar 12 2011

A086700 Euler phi function applied to the triangular numbers.

Original entry on oeis.org

1, 2, 2, 4, 8, 12, 12, 12, 24, 40, 20, 24, 72, 48, 32, 64, 96, 108, 72, 48, 120, 220, 88, 80, 240, 216, 108, 168, 224, 240, 240, 160, 320, 384, 144, 216, 648, 432, 192, 320, 480, 504, 420, 240, 528, 1012, 368, 336, 840, 640, 384, 624, 936, 720, 480, 432, 1008, 1624, 464, 480, 1800, 1080, 576, 768, 960, 1320

Offset: 1

Jon Perry, Jul 28 2003

			a(3) = phi(6) = 2.

Michel Marcus, Table of n, a(n) for n = 1..5000

Cf. A000010, A000217, A065474.

with(numtheory):with(combinat):a:=n->phi(binomial(n,2)): seq(a(n), n=2..31); # Zerinvary Lajos, Oct 05 2007

EulerPhi[Accumulate[Range[70]]] (* Harvey P. Dale, Sep 16 2012 *)

PARI
```
vector(66,n,eulerphi(n*(n+1)/2))
```

[euler_phi(binomial(n,2)) for n in range(2,32)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000010(A000217(n)). - Michel Marcus, Aug 21 2017

Sum_{k=1..n} a(k) = c * n^3 / 4 + O((n*log(n))^2), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 09 2024

A087618 a(n) is the number of pair of consecutive numbers (k,k+1) with k<=10^n such that k and k+1 are both squarefree.

Original entry on oeis.org

1, 5, 33, 323, 3230, 32269, 322619, 3226343, 32263377, 322634281, 3226340896, 32263409594, 322634100659, 3226340989192

Offset: 0

Eric W. Weisstein, Sep 19 2003

Eric Weisstein's World of Mathematics, Squarefree

Cf. A071172, A065474.

Asymptotic density is A065474.

a(11)-a(13) from Donovan Johnson, Jun 26 2010

A173186 Numbers k such that k^2-1 is not squarefree.

Original entry on oeis.org

1, 3, 5, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 55, 57, 59, 61, 62, 63, 64, 65, 67, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 87, 89, 91, 93, 95, 97, 98, 99, 100, 101, 103, 105

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 12 2010

nonn

Complement of A067874 - R. J. Mathar, Feb 13 2010

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 2/p^2) = 0.677365... (1 - A065474). - Amiram Eldar, Feb 25 2021

			5 is a term since 5^2-1 = 24 = 2^3*3 is not squarefree.

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

Cf. A013929, A065474, A067874.

f[n_]:=SquareFreeQ[n]; lst={};Do[If[f[n^2-1],AppendTo[lst,n]],{n,6!}]; lst
Select[Range[120],!SquareFreeQ[#^2-1]&] (* Harvey P. Dale, Jul 16 2016 *)

Definition edited by R. J. Mathar, Feb 13 2010

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

Original entry on oeis.org

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

Zak Seidov, Aug 22 2012

nonn

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

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

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

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.

A007674 is a subsequence.

Cf. A000217, A005117, A061304, A065474.

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

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

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

A319210 a(n) = phi(n^2 - 1)/2 where phi is A000010.

Original entry on oeis.org

1, 2, 4, 4, 12, 8, 18, 16, 30, 16, 60, 24, 48, 48, 64, 48, 144, 48, 108, 80, 132, 80, 220, 96, 180, 144, 252, 96, 420, 128, 300, 256, 240, 192, 432, 216, 432, 288, 480, 192, 840, 240, 504, 440, 552, 352, 966, 320, 672, 480, 832, 432, 1040, 432, 720, 672, 1044, 448

Offset: 2

Seiichi Manyama, Sep 13 2018

Seiichi Manyama, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Row 2 of A369291.
Cf. A000010, A005563 (n^2-1, shifted), A065474.
phi(n^b - 1)/b: this sequence (b=2), A319213 (b=3), A319214 (b=5).

[EulerPhi(n^2-1)/2: n in [2..70]]; // Vincenzo Librandi, Sep 15 2018

Table[(EulerPhi@(n^2 - 1) / 2), {n, 2, 70}] (* Vincenzo Librandi, Sep 15 2018 *)

PARI
```
{a(n) = eulerphi(n^2-1)/2}
```

Sum_{k=1..n} a(k) = c * n^3 / 4 + O((n*log(n))^2), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 09 2024

A330319 a(n) = Sum_{i=1..n} phi(i)*phi(i+1), where phi(n) = A000010(n) is Euler's totient function.

Original entry on oeis.org

1, 3, 7, 15, 23, 35, 59, 83, 107, 147, 187, 235, 307, 355, 419, 547, 643, 751, 895, 991, 1111, 1331, 1507, 1667, 1907, 2123, 2339, 2675, 2899, 3139, 3619, 3939, 4259, 4643, 4931, 5363, 6011, 6443, 6827, 7467, 7947, 8451, 9291, 9771, 10299, 11311, 12047, 12719, 13559, 14199, 14967, 16215, 17151, 17871, 18831

Offset: 1

N. J. A. Sloane, Dec 11 2019

nonn

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 32.

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.
L. Mirsky, Summation formula involving arithmetic functions, Duke Mathematical Journal, Vol. 16, No. 2 (1949), pp. 261-272.

Cf. A000010, A065474, A335131.

Partial sums of A083542.

phi = EulerPhi[Range[56]]; Accumulate[Most[phi] * Rest[phi]] (* Amiram Eldar, Mar 05 2020 *)

a(n) = sum(i=1, n, eulerphi(i)*eulerphi(i+1)); \\ Michel Marcus, Mar 05 2020

a(n) ~ (c/3) * n^3 + O(n^2*log(n)^2), where c = Product_{p prime}(1 - 2/p^2) (A065474). - Amiram Eldar, Mar 05 2020

A340153 Decimal expansion of Product_{p prime} (1 - 2/p^3).

Original entry on oeis.org

6, 7, 6, 8, 9, 2, 7, 3, 7, 0, 0, 9, 8, 8, 1, 9, 9, 3, 6, 1, 0, 2, 3, 7, 3, 2, 6, 7, 2, 4, 3, 8, 9, 2, 1, 2, 7, 9, 7, 6, 7, 8, 3, 9, 7, 4, 5, 9, 7, 8, 8, 8, 4, 5, 2, 7, 3, 2, 9, 7, 8, 2, 3, 0, 4, 4, 3, 2, 6, 3, 2, 0, 4, 6, 0, 3, 5, 7, 8, 6, 0, 5, 1, 2, 8, 3, 2, 6, 8, 4, 8, 1, 1, 1, 1, 0, 8, 4, 4, 9, 3, 1, 7, 0, 8, 4

Offset: 0

Amiram Eldar, Dec 29 2020

The asymptotic density of the sequence of cubefree numbers k such that k+1 is also cubefree (A340152) (Carlitz, 1932).

			0.67689273700988199361023732672438921279767839745978...

Leonard Carlitz, On a problem in additive arithmetic (II), The Quarterly Journal of Mathematics, Vol. os-3, No. 1 (1932), pp. 273-290.

Cf. A065474, A340152.

$MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{0, 0, 2}, {0, 0, -6}, m]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n])/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

PARI
```
prodeulerrat(1 - 2/p^3)
```

More digits from Vaclav Kotesovec, Jan 16 2021

A368249 a(n) = A002378(A005117(n)-1).

Original entry on oeis.org

0, 2, 6, 20, 30, 42, 90, 110, 156, 182, 210, 272, 342, 420, 462, 506, 650, 812, 870, 930, 1056, 1122, 1190, 1332, 1406, 1482, 1640, 1722, 1806, 2070, 2162, 2550, 2756, 2970, 3192, 3306, 3422, 3660, 3782, 4160, 4290, 4422, 4692, 4830, 4970, 5256, 5402, 5852, 6006

Offset: 1

Amiram Eldar, Dec 19 2023

The squarefree oblong numbers (A229882) are all terms of this sequence, and their relative asymptotic density in it is A065474/A059956 = 0.530711... (A065469).

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

Cf. A002378, A005117, A229882.

Cf. A059956, A065469, A065474, A368250.

Table[n*(n - 1), {n, Select[Range[100], SquareFreeQ]}]

lista(kmax) = forsquarefree(k=1, kmax, print1(k[1]*(k[1]-1), ", "));

from math import isqrt
from sympy import mobius
def A368249(n):
    def f(x): return int(n-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m*(m-1) # Chai Wah Wu, Dec 23 2024

Sum_{n>=2} 1/a(n) = Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) = 0.848633... (A368250).

A064148 Numbers k such that mu(k) = mu(k+1), where mu is the Möbius function (A008683).

Original entry on oeis.org

2, 8, 14, 21, 24, 27, 29, 30, 33, 34, 38, 41, 42, 44, 48, 49, 57, 63, 66, 70, 75, 78, 80, 85, 86, 93, 94, 98, 99, 101, 102, 109, 113, 116, 118, 120, 122, 124, 125, 130, 133, 135, 137, 138, 141, 142, 145, 147, 152, 158, 168, 171, 173, 175, 177, 181, 188, 190, 201

Offset: 1

Jason Earls, Sep 11 2001

			2 is a term since mu(2) = mu(3) = -1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A008683, A074820, A065474.

SequencePosition[MoebiusMu[Range[250]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 26 2017 *)

j=[]; for(n=1,500, if(moebius(n)==moebius(n+1),j=concat(j,n))); j

{ n=0; for (m=1, 10^9, if (moebius(m)==moebius(m + 1), write("b064148.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 09 2009

a(n) seems to be asymptotic to c*n with c=3.7.... Using heuristic arguments (cf. A074820): c maybe = 1/(3A/2 + 1 - 12/Pi^2) ~ 3.729994018, where A ~ 0.3226340989 is the product over all primes p of 1 - 2/p^2 (cf. A065474). - Benoit Cloitre, Sep 08 2002

cp's OEIS Frontend

A117204 Squarefree positive integers k such that 2*k+1 is also squarefree.

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A086700 Euler phi function applied to the triangular numbers.

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Maple

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A087618 a(n) is the number of pair of consecutive numbers (k,k+1) with k<=10^n such that k and k+1 are both squarefree.

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A173186 Numbers k such that k^2-1 is not squarefree.

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Mathematica

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

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A319210 a(n) = phi(n^2 - 1)/2 where phi is A000010.

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Magma

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A330319 a(n) = Sum_{i=1..n} phi(i)*phi(i+1), where phi(n) = A000010(n) is Euler's totient function.

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