A065473 -id:A065473 - cp's OEIS frontend

A256390 a(n) = number of triples (a,b,c) of natural numbers a,b,c <= n with gcd(a,b)=gcd(b,c)=gcd(c,a)=1.

1, 4, 13, 22, 55, 64, 133, 172, 247, 280, 469, 508, 781, 868, 997, 1144, 1621, 1714, 2323, 2488, 2785, 3010, 3907, 4078, 4837, 5176, 5833, 6178, 7627, 7798, 9463, 10102, 10927, 11530, 12631, 13006, 15379, 16150, 17311, 17926, 20863, 21256

Offset: 1

Juan Arias-de-Reyna, Mar 27 2015

nonn

The sequence has asymptotics rho*n^3+O(n^2 log^2n) with rho=prod_p(1-3/p^2+2/p^3)=0.2867474284344...(product on primes). See A065473.

			a(3)=13 because the 13 triples (1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,1,3), (1,3,1), (3,1,1), (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..1399
J. Arias de Reyna and R. Heyman, Counting tuples restricted by pairwise primality, arXiv:1403.2769 [math.NT], 2014.
J. Arias de Reyna, R. Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq. 18 (2015) 15.10.4

Cf. A256391.

A[M_] := A[M] = Module[{X, a1, a2, a3, K, count, k},
    X = Flatten[
      Table[{a1, a2, a3}, {a1, 1, M}, {a2, 1, M}, {a3, 1, M}], 2];
    K = Length[X];
    count = 0;
    For[k = 1, k <= K, k++,
     {a1, a2, a3} = X[[k]];
     If[(GCD[a1, a2] == 1) && (GCD[a2, a3] == 1) && (GCD[a3, a1] ==
         1), count = count + 1]];
    count];
Table[A[n], {n, 1, 100}]

a(n) = sum_a sum_b sum_c mu(a) mu(b) mu(c) [n/gcd(a,b)][n/gcd(b,c)][n/gcd(c,a)], where mu(.) is Moebius function [x] integer part of x, and a,b,c run through natural numbers.

A319592 Decimal expansion of the probability that an integer 4-tuple is pairwise coprime.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 27 2019

			0.114884044080228788729251276701599097848713552687283...

László Tóth, The probability that k positive integers are pairwise relatively prime, Fibonacci Quarterly, Vol. 40, No. 1 (2002), pp. 13-18.

Cf. A059956, A065473, A256392.

$MaxExtraPrecision = 1000; nm = 1000; c = LinearRecurrence[{-2, 3}, {0, -12}, nm]; f[x_] := (1 - x)^3*(1 + 3*x); RealDigits[f[1/2]*f[1/3]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k - 1/3^k)/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]

prodeulerrat((1 - 1/p)^3 * (1 + 3/p)) \\ Amiram Eldar, Jun 29 2023

Equals Product_{p prime} (1 - 1/p)^3 * (1 + 3/p).

A015632 Number of ordered triples of integers from [ 1,n ] with no common factors between pairs.

Original entry on oeis.org

1, 2, 4, 6, 12, 14, 26, 33, 46, 52, 84, 91, 137, 152, 174, 199, 279, 295, 397, 425, 475, 513, 663, 692, 819, 876, 986, 1044, 1286, 1315, 1593, 1700, 1838, 1939, 2123, 2186, 2582, 2711, 2905, 3008, 3498, 3564, 4106, 4272, 4476, 4666, 5316, 5433, 5985

Offset: 1

Olivier Gérard

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..500

a015632 n = length [(x,y,z) | z <- [1..n], y <- [1..z], gcd y z == 1,
                              x <- [1..y], gcd x z == 1, gcd x y == 1]
-- Reinhard Zumkeller, Nov 14 2011

a(n) ~ k * n^3 where k is 1/6 * A065473 = 0.04779.... [Charles R Greathouse IV, Nov 14 2011]

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

A384531 Multiplicative sequence a(n) with a(p^e) = ((2e+1) p - 2e) p^(e-1) for prime p and e >= 0.

Original entry on oeis.org

1, 4, 7, 12, 13, 28, 19, 32, 33, 52, 31, 84, 37, 76, 91, 80, 49, 132, 55, 156, 133, 124, 67, 224, 105, 148, 135, 228, 85, 364, 91, 192, 217, 196, 247, 396, 109, 220, 259, 416, 121, 532, 127, 372, 429, 268, 139, 560, 217, 420, 343, 444, 157, 540, 403, 608, 385, 340, 175, 1092

Offset: 1

Werner Schulte, Jun 01 2025

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
David Radcliffe, Notes on two arithmetical functions (Contains a proof that a(n) = Sum_{i=1..n} gcd(i, n) * gcd(i+1, n)).

Cf. A018804, A173557.

Cf. A065473.

A384531 := proc(n)
    local a,pe,p,e;
    a :=1 ;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        a := a*((2*e+1) * p - 2*e) * p^(e-1) ;
    end do:
    a ;
end proc:
seq(A384531(n),n=1..100) ;# R. J. Mathar, Jun 04 2025

f[p_, e_] := ((2*e+1)*p - 2*e)*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 01 2025 *)

a(n)=my(f=factor(n)); prod(k=1,#f[,1],((2*f[k,2]+1)*f[k,1]-2*f[k,2])*f[k,1]^(f[k,2]-1))

from math import prod
from sympy import factorint
def A384531(n): return prod((((m:=e<<1)|1)*p-m)*p**(e-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 03 2025

Dirichlet g.f.: Sum_{n > 0} a(n) / n^s = (zeta(s-1))^2 * Product_{p prime} (1 + (p-2) / p^s).
Dirichlet convolution of A018804 and A173557.
Conjecture: a(n) = Sum_{i=1..n} gcd(i, n) * gcd(i+1, n).
From Vaclav Kotesovec, Jun 04 2025: (Start)
Let f(s) = Product_{primes p} (1 + 2/p^(2*s-1) - 1/p^(2*s-2) - 2/p^s).
Dirichlet g.f.: zeta(s-1)^3 * f(s).
Sum_{k=1..n} a(k) ~ f(2) * n^2 * (log(n)^2 + (6*gamma - 1 + 2*f'(2)/f(2))*log(n) + 1/2 - 3*gamma + 6*gamma^2 - 6*sg1 + (6*gamma - 1)*f'(2)/f(2) + f''(2)/f(2))/4, where
f(2) = Product_{primes p} (1 - 3/p^2 + 2/p^3) = A065473 = 0.2867474284344787341...,
f'(2) = f(2) * Sum_{primes p} 4*log(p)/(p^2 + p - 2) = 0.53488225650873164189786660885838556843579696135554271633442328...,
f''(2) = f'(2)^2/f(2) + f(2) * Sum_{primes p} (-2*p*(3*p+2)*log(p)^2 / (p^2+p-2)^2) = -0.29112624105319980992840485620511000074444413707069816872854442...,
gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

A273093 Decimal expansion of the probability that three positive integers are pairwise not coprime.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 15 2016

			0.1742197830347247005585740721805346916511057518703135572332637051646...

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

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 15.
Randell Heyman, Pairwise non-coprimality of triples, arXiv preprint, arXiv:1309.5578 [math.NT], 2013-2014.
Randell Heyman, Topics in divisibility: pairwise coprimality, the GCD of shifted sets and polynomial irreducibility, PhD thesis, University of New South Wales, 2015.
Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.
Eric Weisstein's MathWorld, Carefree Couple.

Cf. A065464, A065473, A229099.

$MaxExtraPrecision = 1000; digits = 100; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms + 10]]; r[n_Integer] := LR[[n]];
P = (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n - 1]/(n - 1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]];
Q = NSum[-(2 + (-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 3digits, Method -> "AlternatingSigns"]//Exp;
F3 = 1 - 18/Pi^2 + 3P - Q;
RealDigits[F3, 10, digits][[1]]

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

Equals 1 - 18/Pi^2 + 3P - Q, where P is A065464 and Q is A065473.

A319593 Decimal expansion of the probability that an integer triple is pairwise unitary coprime.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 27 2019

Two numbers are unitary coprime if their largest common unitary divisor is 1.

			0.552306904157942811183227347309264708535455831404497...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 54.

László Tóth, Multiplicative arithmetic functions of several variables: a survey, in Themistocles M. Rassias and Panos M. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, NY, 2014, pp. 483-514 (see p. 509), preprint, arXiv:1310.7053 [math.NT], 2013-2014 (see p. 22).

Cf. A065473, A077610, A306071.

$MaxExtraPrecision = 1000; nm = 1000; f[x_] := 1 - 4*x^2 + 7*x^3 - 9*x^4 + 8*x^5 - 2*x^6 - 3*x^7 + 2*x^8; c = LinearRecurrence[{-1, 3, -4, 5, -3, -1, 2}, {0, -8, 21, -68, 180, -503, 1428}, nm]; RealDigits[f[1/2] * f[1/3] * Zeta[2] * Zeta[3] * Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k - 1/3^k)/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]

zeta(2) * zeta(3) * prodeulerrat(1-4/p^2+7/p^3-9/p^4+8/p^5-2/p^6-3/p^7+2/p^8) \\ Amiram Eldar, Jun 29 2023

Equals zeta(2) * zeta(3) * Product_{p prime} (1 - 4/p^2 + 7/p^3 - 9/p^4 + 8/p^5 - 2/p^6 - 3/p^7 + 2/p^8).

cp's OEIS Frontend

A256390 a(n) = number of triples (a,b,c) of natural numbers a,b,c <= n with gcd(a,b)=gcd(b,c)=gcd(c,a)=1.

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A319592 Decimal expansion of the probability that an integer 4-tuple is pairwise coprime.

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A015632 Number of ordered triples of integers from [ 1,n ] with no common factors between pairs.

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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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A384531 Multiplicative sequence a(n) with a(p^e) = ((2*e+1) * p - 2*e) * p^(e-1) for prime p and e >= 0.

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A273093 Decimal expansion of the probability that three positive integers are pairwise not coprime.

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A319593 Decimal expansion of the probability that an integer triple is pairwise unitary coprime.

A384531 Multiplicative sequence a(n) with a(p^e) = ((2e+1) p - 2e) p^(e-1) for prime p and e >= 0.