A065473 -id:A065473 - cp's OEIS frontend

A015617 Number of (unordered) triples of integers from [1,n] with no common factors between pairs.

0, 0, 1, 2, 7, 8, 19, 25, 37, 42, 73, 79, 124, 138, 159, 183, 262, 277, 378, 405, 454, 491, 640, 668, 794, 850, 959, 1016, 1257, 1285, 1562, 1668, 1805, 1905, 2088, 2150, 2545, 2673, 2866, 2968, 3457, 3522, 4063, 4228, 4431, 4620, 5269, 5385, 5936

Offset: 1

Olivier Gérard

nonn

Form the graph with nodes 1..n, joining two nodes by an edge if they are relatively prime; a(n) = number of triangles in this graph. - N. J. A. Sloane, Feb 06 2011. The number of edges in this graph is A015614. - Roberto Bosch Cabrera, Feb 07 2011.

			For n=5, there are a(5)=7 triples: (1,2,3), (1,2,5), (1,3,4), (1,3,5), (1,4,5), (2,3,5) and (3,4,5) out of binomial(5,3) = 10 triples of distinct integers <= 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Subset of A015616 (triples with no common factor) and A015631 (ordered triples with no common factor).

Cf. A185953 (first differences), A186230, Column 3 of triangle A186974.

a[n_] := Select[Subsets[Range[n], {3}], And @@ (GCD @@ # == 1 & /@ Subsets[#, {2}]) &] // Length; a /@ Range[49]
(* Jean-François Alcover, Jul 11 2011 *)

a(n)=sum(a=1,n-2,sum(b=a+1,n-1,sum(c=b+1,n, gcd(a,b)==1 && gcd(a,c)==1 && gcd(b,c)==1))) \\ Charles R Greathouse IV, Apr 28 2015

For large n one can show that a(n) ~ C*binomial(n,3), where C = 0.28674... = A065473. - N. J. A. Sloane, Feb 06 2011.

a(n) = Sum_{r=1..n} Sum_{k=1..r} A186230(r,k). - Alois P. Heinz, Feb 17 2011

Added one example and 2 cross-references. - Olivier Gérard, Feb 06 2011.

A069212 a(n) = Sum_{k=1..n} 3^omega(k).

Original entry on oeis.org

1, 4, 7, 10, 13, 22, 25, 28, 31, 40, 43, 52, 55, 64, 73, 76, 79, 88, 91, 100, 109, 118, 121, 130, 133, 142, 145, 154, 157, 184, 187, 190, 199, 208, 217, 226, 229, 238, 247, 256, 259, 286, 289, 298, 307, 316, 319, 328, 331, 340, 349, 358, 361, 370, 379, 388, 397

Offset: 1

Benoit Cloitre, Apr 14 2002

More generally, if b is an integer =>3, Sum_{k=1..n} b^omega(k) ~ C(b)*n*log(n)^(b-1) where C(b)=1/(b-1)!*prod((1-1/p)^(b-1)*(1+(b-1)/p)).

G. Tenenbaum and Jie Wu, Cours Spécialisés No. 2: "Théorie analytique et probabiliste des nombres", Collection SMF, Ordres moyens, p. 20.
G. Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., American Mathematical Soc. (2015). See page 59.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Partial sums of A074816.

Cf. A001222, A064608, A065473, A350961.

Accumulate @ Table[3^PrimeNu[n], {n, 1, 57}] (* Amiram Eldar, May 24 2020 *)

from sympy.ntheory.factor_ import primenu
def A069212(n): return sum(3**primenu(m) for m in range(1,n+1)) # Chai Wah Wu, Sep 07 2023

Asymptotic formula: a(n) ~ C*n*log(n)^2 with C = (1/2) * Product_{p} ((1-1/p)^2*(1+2/p)) where the product is over all the primes.
The constant C is A065473/2. - Amiram Eldar, May 24 2020
From Ridouane Oudra, Jan 01 2021: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..n} mu(i*j)^2*floor(n/(i*j));
a(n) = Sum_{i=1..n} mu(i)^2*tau(i)*floor(n/i);
a(n) = Sum_{i=1..n} 2^Omega(i)*mu(i)^2*floor(n/i), where Omega = A001222. (End)
From Vaclav Kotesovec, Feb 16 2022: (Start)
More precise asymptotics:
Let f(s) = Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)), then
a(n) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2),
where f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.2867474284344787341078927127898384464343318440970569956414778593366522...,
f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = 0.8023233847630974628467999132875783526536954420333140745016349208975965...,
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} -6*p*(2*p+1) * log(p)^2 / (p^2 + p - 2)^2 = -0.255987592484328884627082229528266165335336670389046663124468278519...
and gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

A299822 Product of Euler's totient and the squarefree kernel, a(n) = phi(n)*rad(n).

Original entry on oeis.org

1, 2, 6, 4, 20, 12, 42, 8, 18, 40, 110, 24, 156, 84, 120, 16, 272, 36, 342, 80, 252, 220, 506, 48, 100, 312, 54, 168, 812, 240, 930, 32, 660, 544, 840, 72, 1332, 684, 936, 160, 1640, 504, 1806, 440, 360, 1012, 2162, 96, 294, 200, 1632, 624, 2756, 108, 2200, 336, 2052, 1624

Offset: 1

R. J. Mathar, Feb 19 2018

A permutation of A323333. - Amiram Eldar, Sep 19 2020

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

Cf. A000010, A007947, A065485, A173557, A323333.

A299822 := proc(n)
    local a,p,e,pe;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ; e:= pe[2] ;
        a := a*p*(p-1)*p^(e-1) ;
    end do:
    a ;
end proc:
seq(A299822(n),n=1..130) ;

Array[EulerPhi[#] SelectFirst[Reverse@ Divisors@ #, SquareFreeQ] &, 58] (* Michael De Vlieger, Feb 20 2018 *)
f[p_, e_] := (p-1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n) = eulerphi(n)*factorback(factorint(n)[, 1]); \\ Michel Marcus, Jun 24 2019

a(n) = A000010(n)*A007947(n) = n*A173557(n).
Dirichlet g.f.: zeta(s-1)*Product_{p prime} (1 - 2*p^(1-s) + p^(2-s)), corrected by Vaclav Kotesovec, Dec 18 2019
Multiplicative with a(p^e) = p*(p-1)*p^(e-1).
a(n) = n*abs(A023900(n)). (Trivially rephrasing a formula in A173557.) - Omar E. Pol, Feb 19 2018
a(2^e) = 2^e. (Special case of above.) - Omar E. Pol, Feb 19 2018
A003557(n) | a(n). - R. J. Mathar, Feb 26 2018
From Vaclav Kotesovec, Dec 18 2019: (Start)
Dirichlet g.f.: zeta(s-1) * zeta(s-2) * Product_{primes p} (1 + 2*p^(3-2*s) - p^(4-2*s) - 2*p^(1-s)).
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^3 / 18, where c = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.2867474284344787341078927... (End)
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p-1)^2) = 2.826419... (A065485). - Amiram Eldar, Sep 19 2020
G.f. for a signed version of the sequence: Sum_{n >= 1} mu(n)*n^2*x^n/(1 - x^n)^2 = Sum_{n >= 1} (-1)^omega(n)*a(n)*x^n = x - 2*x^2 - 6*x^3 - 4*x^4 - 20*x^5 + 12*x^6 - 42*x^7 - 8*x^8 - 18*x^9 + 40*x^10 - ..., where mu(n) is the Möbius function A008683(n) and omega(n) = A001221(n) is the number of distinct primes dividing n. - Peter Bala, Mar 05 2022

A330594 Decimal expansion of Product_{primes p} (1 + 1/p^2 - 2/p^3).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Dec 19 2019

			1.106960111953217676651179130007439592949548833658122419043134044978777...

Cf. A065464, A065473, A065476, A208133, A328017, A330595, A330596.

Do[Print[N[Exp[-Sum[q = Expand[(-p^2 + 2*p^3)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 110]], {t, 20, 200, 20}]

prodeulerrat(1 + 1/p^2 - 2/p^3) \\ Amiram Eldar, Mar 16 2021

A065480 Decimal expansion of Product_{p prime} (1 - 1/(p^2+p-1)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

The probability that two numbers are coprime given that they are both coprime to a randomly chosen third number. - Luke Palmer, Apr 27 2019

			0.6695802905390623676350256956124342...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A065592, A078081.

Cf. A013661, A065463, A065473.

digits = 98; Exp[NSum[(1/2)*(-2 + (-2)^n - ((1/2)*(-1 - Sqrt[5]))^n*(-1 + Sqrt[5]) + ((1/2)*(-1 + Sqrt[5]))^n*(1 + Sqrt[5]))*PrimeZetaP[n - 1]/(n - 1), {n, 3, Infinity}, WorkingPrecision -> 4 digits, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 1/(p^2+p-1)) \\ Amiram Eldar, Mar 14 2021

Equals A065473*zeta(2)/A065463. - Luke Palmer, Apr 27 2019

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

Original entry on oeis.org

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)

A069201 a(n) = Sum_{k=1..n} mu(k)^2 * 2^omega(k) where omega(k) is the number of distinct primes in the factorization of k.

Original entry on oeis.org

1, 3, 5, 5, 7, 11, 13, 13, 13, 17, 19, 19, 21, 25, 29, 29, 31, 31, 33, 33, 37, 41, 43, 43, 43, 47, 47, 47, 49, 57, 59, 59, 63, 67, 71, 71, 73, 77, 81, 81, 83, 91, 93, 93, 93, 97, 99, 99, 99, 99, 103, 103, 105, 105, 109, 109, 113, 117, 119, 119, 121, 125, 125, 125, 129, 137

Offset: 1

Benoit Cloitre, Apr 14 2002

G. Tenenbaum and Jie Wu, Cours Spécialisés No. 2: "Théorie analytique et probabiliste des nombres", Collection SMF, Ordres moyens, p. 20.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Partial sums of A074823.

Cf. A008966, A034444, A065473, A347149.

[&+[MoebiusMu(k)^2*#Divisors(k):k in [1..n]]: n in [1..66]]; // Marius A. Burtea, Jul 27 2019

with(numtheory): seq(add(tau(k)*mobius(k)^2, k=1..n), n=1..90); # Ridouane Oudra, Jul 25 2019

Accumulate @ Table[MoebiusMu[n]^2 * 2^PrimeNu[n], {n, 1, 66}] (* Amiram Eldar, May 24 2020 *)

a(n) = sum(k=1, n, moebius(k)^2*2^omega(k)); \\ Michel Marcus, Jul 23 2017

(define (A069201 n) (if (= 1 n) n (+ (A074823 n) (A069201 (- n 1))))) ;; Antti Karttunen, Jul 23 2017

Asymptotic formula: a(n) = C*n*log(n) + O(n) with C = Product_{p prime} (1 - 1/p)^2*(1 + 2/p).
The constant C is A065473. - Amiram Eldar, May 24 2020
a(n) = Sum_{k=1..n} mu(k)^2*d(k), where d is the number of divisors function (A000005). - Ridouane Oudra, Jul 25 2019
More precise asymptotics: Let f(s) = Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)), then a(n) ~ n*(f(1)*(log(n) + 2*gamma - 1) + f'(1)), where f(1) = A065473, f'(1) = f(1) * Sum_{primes p} 6*log(p)/(p^2 + p - 2) = 0.802323384763097462846799913287578352653695442033314074501634920897596526... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 20 2021

A158274 Numerators of antiharmonic means of divisors of n.

Original entry on oeis.org

1, 5, 5, 3, 13, 25, 25, 17, 7, 65, 61, 15, 85, 125, 65, 11, 145, 35, 181, 13, 125, 305, 265, 85, 21, 425, 41, 75, 421, 325, 481, 65, 305, 725, 325, 21, 685, 181, 425, 221, 841, 625, 925, 61, 91, 1325, 1105, 55, 43, 35

Offset: 1

Jaroslav Krizek, Mar 15 2009

Numbers k such that sigma_2(k)/sigma_1(k) = A001157(k)/A000203(k) are integers are in A020487.

			Antiharmonic means of divisors of n>=1: 1, 5/3, 5/2, 3, 13/2, 25/6, ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A001157, A020487, A065473, A152649, A158275 (denominators)

Table[Numerator[DivisorSigma[2, n]/DivisorSigma[1, n]], {n, 50}] (* Ivan Neretin, May 22 2015 *)

a(n) = numerator(sigma(n,2)/sigma(n)); \\ Amiram Eldar, Nov 21 2022

Antiharmonic mean of divisors of number n = Product (p_i^e_i) is sigma_2(n)/sigma_1(n) = A001157(n)/A000203(n) = Product ((p_i^(e_i+1)+1)/(p_i+1)).

Sum_{k=1..n} a(k)/A158275(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - (3*p-2)/(p^3)) = A152649 * A065473 = 0.387941... . - Amiram Eldar, Nov 21 2022

A061780 Number of solutions to x + y + z = 0 mod (2n+1) such that x,y,z are units modulo 2n+1, i.e., gcd(x, 2n+1) = gcd(y, 2n+1) = gcd(z, 2n+1) = 1.

Original entry on oeis.org

2, 12, 30, 18, 90, 132, 24, 240, 306, 60, 462, 300, 162, 756, 870, 180, 360, 1260, 264, 1560, 1722, 216, 2070, 1470, 480, 2652, 1080, 612, 3306, 3540, 540, 1584, 4290, 924, 4830, 5112, 600, 2700, 6006, 1458, 6642, 2880, 1512, 7656, 3960, 1740, 3672, 9120

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 22 2001

This sequence is not multiplicative. What is multiplicative is the sequence b = 1,0,2,0,12,0,30, ... such that a(n) = b(2n+1) and b(2n)=0. - Robert Israel, Jan 29 2017

			The only solutions modulo 3 in units are 1+1+1 = 0 mod 3, 2+2+2 = 0 mod 3 so the first element of the sequence is 2.

Robert Israel, Table of n, a(n) for n = 1..10000
Masao Arai and Jiyu Gakuen, Problem E 1460, American Mathematical Monthly, Vol. 68, No. 3 (1961), p. 295 and A Number-Theoretic Function, solution by Leonard Carlitz, American Mathematical Monthly, Vol. 68, No. 9 (1961), pp. 932-933.
László Tóth, Another generalization of Euler's arithmetic function and Menon's identity, The Ramanujan Journal (2021).

Cf. A065473.

f:= n -> n^2*mul((1-1/p)*(1-2/p),p=numtheory:-factorset(n)):
seq(f(2*n+1),n=1..100); # Robert Israel, Jan 29 2017

a[n_] := (2*n+1)^2 * Product[(1-1/p)*(1-2/p), {p, FactorInteger[2*n+1][[;;,1]]}]; Array[a, 50] (* Amiram Eldar, Jan 03 2022 *)

If 2n+1 = p^k is a prime power with p an odd prime then a(n) = p^(2k-2) * (p^2 - 3p + 2).
a(n) = (2n+1)^2 * Product_{primes p | 2n+1} (1 - 3/p + 2/p^2). - Robert Israel, Jan 29 2017
Sum_{k=1..n} a(k) ~ c * (2*n)^3/3 + O(n^2*log(n)^3), where c = A065473 (Tóth, 2021). - Amiram Eldar, Jan 03 2022

More terms from Vladeta Jovovic, Jun 23 2001

Corrected by Robert Israel, Jan 29 2017

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

cp's OEIS Frontend

A015617 Number of (unordered) triples of integers from [1,n] with no common factors between pairs.

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A069212 a(n) = Sum_{k=1..n} 3^omega(k).

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A299822 Product of Euler's totient and the squarefree kernel, a(n) = phi(n)*rad(n).

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A330594 Decimal expansion of Product_{primes p} (1 + 1/p^2 - 2/p^3).

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A065480 Decimal expansion of Product_{p prime} (1 - 1/(p^2+p-1)).

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

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A069201 a(n) = Sum_{k=1..n} mu(k)^2 * 2^omega(k) where omega(k) is the number of distinct primes in the factorization of k.

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