A118260 -id:A118260 - cp's OEIS frontend

A118259 Numbers of strongly carefree couples (a,b) with a,b <= n.

1, 3, 7, 7, 13, 17, 27, 27, 27, 33, 47, 47, 63, 73, 85, 85, 107, 107, 131, 131, 147, 165, 195, 195, 195, 215, 215, 215, 249, 265, 303, 303, 329, 355, 385, 385, 431, 461, 495, 495, 547, 569, 625, 625, 625, 661, 721, 721, 721, 721, 763, 763, 827, 827, 877, 877

Offset: 1

Eric W. Weisstein, Apr 20 2006

nonn

(a, b) is a strongly carefree couple if gcd(a, b) = 1 and both a and b are 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, A065473, A118258, A118260.

Table[nn = n;Length[Select[Level[Table[Table[{i, j}, {i, 1, nn}], {j, 1, nn}], {2}],Apply[GCD, #] == 1 && SquareFreeQ[#[[1]]] &&SquareFreeQ[#[[2]]] &]], {n, 1, 56}] (* Geoffrey Critzer, Jan 13 2015 *)

a(n)=sum(i=1,n,sum(j=1,n,moebius(i*j)^2)) \\ Benoit Cloitre, Oct 10 2009

a(n) = Sum_{i,j=1...n} mu(i*j)^2. - Benoit Cloitre, Oct 10 2009
From Amiram Eldar, Mar 03 2021: (Start)
a(n) = 2*A118258(n) - A118260(n).
a(n) ~ A065473 * n^2 + O(n*log(n)). (End)

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

Original entry on oeis.org

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)

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.

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A118259 Numbers of strongly carefree couples (a,b) with a,b <= n.

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

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A118261 Decimal expansion of probability of a weakly carefree couple.

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Mathematica

PARI

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