A118259 -id:A118259 - cp's OEIS frontend

A118258 Numbers of carefree couples (a,b) with a,b<=n.

1, 3, 7, 9, 16, 20, 31, 35, 39, 46, 63, 67, 87, 98, 112, 119, 146, 152, 182, 189, 209, 228, 265, 273, 286, 308, 321, 330, 375, 391, 440, 453, 486, 515, 554, 565, 624, 657, 698, 712, 778, 801, 871, 888, 906, 946, 1022, 1037, 1063, 1080, 1133, 1152, 1236, 1252

Offset: 1

Eric W. Weisstein, Apr 20 2006

nonn

(a, b) is a carefree couple if gcd(a, b) = 1 and a is squarefree (A005117). - Amiram Eldar, Mar 03 2021

			a(4) = 9 because there are 9 ordered pairs (i,j) of positive integers such that 1<=i,j<=4, gcd(i,j)=1 and i is squarefree: (1,1), (2,1), (3,1), (1,2), (3,2), (1,3), (2,3), (1,4), (3,4). - _Geoffrey Critzer_, Jan 12 2015

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.5.1 Carefree Couples, p. 110.

Charles R Greathouse IV, Table of n, a(n) for n = 1..100000
Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A005117, A065464, A118259, A118260.

F:= proc(n) local A,a;
  A:= select(numtheory:-issqrfree, [$1..n]);
  add(nops(select(y->igcd(a,y)=1, [$1..n])),a=A);
end proc:
seq(F(n),n=1..100); # Robert Israel, Jan 12 2015

Table[nn = n;Select[Level[Table[Table[{i, j}, {i, 1, nn}], {j, 1, nn}], {2}], Apply[GCD, #] == 1 && SquareFreeQ[#[[1]]] &] // Length, {n, 1, 54}] (* Geoffrey Critzer, Jan 12 2015 *)

a(n)=my(s); forsquarefree(m=1,n, s+=sumdiv(m, d, n\d*moebius(d))); s \\ Charles R Greathouse IV, Jan 25 2018

From Amiram Eldar, Mar 03 2021: (Start)
a(n) = (A118259(n) + A118260(n))/2.
a(n) ~ A065464 * n^2 + O(n*log(n)). (End)

A118260 Number of weakly carefree couples (a,b) with a,b<=n.

Original entry on oeis.org

1, 3, 7, 11, 19, 23, 35, 43, 51, 59, 79, 87, 111, 123, 139, 153, 185, 197, 233, 247, 271, 291, 335, 351, 377, 401, 427, 445, 501, 517, 577, 603, 643, 675, 723, 745, 817, 853, 901, 929, 1009, 1033, 1117, 1151, 1187, 1231, 1323, 1353, 1405, 1439, 1503, 1541

Offset: 1

Eric W. Weisstein, Apr 20 2006

nonn

(a, b) is a weakly carefree couple if gcd(a, b) = 1 and either a or b is squarefree (A005117). - Amiram Eldar, Mar 03 2021

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.5.1 Carefree Couples, p. 110.

Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A005117, A118258, A118259, A118261.

From Amiram Eldar, Mar 03 2021: (Start)
a(n) = 2*A118258(n) - A118259(n).
a(n) ~ A118261 * n^2 + O(n*log(n)). (End)

A070072 Number of distinct rectangles with integer sides <= n and squarefree area.

Original entry on oeis.org

1, 2, 4, 4, 7, 9, 14, 14, 14, 17, 24, 24, 32, 37, 43, 43, 54, 54, 66, 66, 74, 83, 98, 98, 98, 108, 108, 108, 125, 133, 152, 152, 165, 178, 193, 193, 216, 231, 248, 248, 274, 285, 313, 313, 313, 331, 361, 361, 361, 361, 382, 382, 414

Offset: 1

Reinhard Zumkeller, Apr 21 2002

nonn

			There are seven rectangles with sides <= 5 having a squarefree area: 1 X 1, 1 X 2, 1 X 3, 1 X 5, 2 X 3, 2 X 5 and 3 X 5, whereas 1 X 4, 2 X 2, 2 X 4, 3 X 3, 3 X 4, 4 X 4, 4 X 5 and 5 X 5 are not squarefree; therefore a(5) = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Squarefree.

Cf. A005117, A065473, A070073.

Cf. A008966, A008683, A118259.

a070072 n = length [() | x <- [1..n], y <- [1..x], a008966 (x*y) == 1]
-- Reinhard Zumkeller, May 26 2012

[&+[&+[MoebiusMu(i*j)^2:j in [1..i]]:i in [1..n]]:n in [1..53]]; // Marius A. Burtea, Oct 17 2019

a(n) = Sum_{i=1..n} Sum_{j= 1..i} mu(i*j)^2, where mu is the Moebius function (A008683). - Ridouane Oudra, Oct 17 2019
a(n) = (A118259(n) + 1)/2. - Ridouane Oudra, May 06 2025
a(n) =  c * n^2 / 2 + O(n*log(n)), where c = Product_{p prime} (1 - (3*p-2)/(p^3)) (A065473). - Amiram Eldar, May 12 2025

A118261 Decimal expansion of probability of a weakly carefree couple.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Apr 20 2006

			0.5697515829197101463296387...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.5.1 Carefree Couples, p. 110.

Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A065464, A065473, A118259, A118260.

$MaxExtraPrecision = 1000; digits = 100; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms + 10]]; r[n_Integer] := LR[[n]];
K1 = (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n - 1]/(n - 1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]];
K2 = NSum[-(2 + (-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 3digits, Method -> "AlternatingSigns"]//Exp;
RealDigits[2 K1 - K2, 10, digits][[1]] (* Jean-François Alcover, May 15 2016 *)

2 * prodeulerrat(1 - (2*p-1)/p^3) - prodeulerrat(1 - (3*p-2)/(p^3)) \\ Amiram Eldar, Mar 03 2021

Equals 2*K1 - K2, where K1 = A065464 and K2 = A065473.

A169646 Number of squarefree numbers of form k*n, 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 0, 3, 7, 0, 8, 5, 6, 0, 11, 0, 12, 0, 8, 9, 15, 0, 0, 10, 0, 0, 17, 8, 19, 0, 13, 13, 15, 0, 23, 15, 17, 0, 26, 11, 28, 0, 0, 18, 30, 0, 0, 0, 21, 0, 32, 0, 25, 0, 23, 23, 36, 0, 37, 25, 0, 0, 30, 18, 41, 0, 29, 22, 44, 0, 45, 30, 0, 0, 36, 22, 49, 0, 0, 32, 51, 0, 41, 34

Offset: 1

Reinhard Zumkeller, Apr 05 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A000040, A008966, A065473, A073311, A112929, A118259.

[&+[MoebiusMu(k*n)^2: k in [1..n]]: n in [1..80]]; // Vincenzo Librandi, Jul 25 2019

with(numtheory): seq(add(mobius(n*i)^2, i = 1 .. n), n = 1 .. 90); # Ridouane Oudra, Jul 24 2019

Count[#,?SquareFreeQ]&/@Table[k*n,{n,90},{k,n}] (* _Harvey P. Dale, Sep 05 2012 *)
Table[Sum[MoebiusMu[i n]^2, {i, n}], {n, 100}] (* Vincenzo Librandi, Jul 25 2019 *)

a(n) = sum(i = 1, n, issquarefree(i*n)); \\ Amiram Eldar, May 12 2025

a(n) = A008966(n)*A073311(n).
a(A000040(n)) = A112929(n).
a(n) = Sum_{i=1..n} A008966(n*i). - Ridouane Oudra, Jul 24 2019
a(n) = (A118259(n) - A118259(n-1))/2, for n>1. - Ridouane Oudra, May 04 2025
Sum_{k=1..n} a(k) ~ c * n / 2, where c = Product_{p prime} (1 - (3*p-2)/(p^3)) (A065473). - Amiram Eldar, May 12 2025

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A118258 Numbers of carefree couples (a,b) with a,b<=n.

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A118260 Number of weakly carefree couples (a,b) with a,b<=n.

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A070072 Number of distinct rectangles with integer sides <= n and squarefree area.

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A118261 Decimal expansion of probability of a weakly carefree couple.

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A169646 Number of squarefree numbers of form k*n, 1 <= k <= n.

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