A065473 -id:A065473 - cp's OEIS frontend

A078073 Continued fraction for constant defined in A065473.

0, 3, 2, 19, 3, 12, 1, 5, 1, 5, 1, 5, 2, 1, 1, 1, 1, 1, 3, 7, 2, 7, 1, 1, 14, 5, 1, 330, 10, 4, 1, 1, 1, 1, 6, 1, 6, 1, 1, 6, 2, 1, 8, 1, 1, 1, 2, 6, 1, 19, 1, 3, 1, 5, 2, 1, 8, 13, 1, 7, 4, 1, 2, 2, 2, 1, 2, 1, 5, 2, 46, 1, 1, 3, 6, 1, 1, 1, 4, 2, 18, 2, 4, 2, 125, 2, 3, 1, 4, 117, 1, 1

Offset: 0

Benoit Cloitre, Dec 02 2002

Cf. A065473 (decimal expansion).

contfrac(prodeulerrat(1 - (3*p-2)/(p^3))) \\ Amiram Eldar, Jul 08 2024

Offset changed by Andrew Howroyd, Jul 05 2024

A065463 Decimal expansion of Product_{p prime} (1 - 1/(p*(p+1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

The density of A268335. - Vladimir Shevelev, Feb 01 2016

The probability that two numbers are coprime given that one of them is coprime to a randomly chosen third number. - Luke Palmer, Apr 27 2019

			0.7044422009991655927366033503...

Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq., Vol. 16 (2013), Article 13.6.3.
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.
David Handelman, Invariants for critical dimension groups and permutation-Hermite equivalence, arXiv preprint arXiv:1309.7417 [math.AC], 2013-2017.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arxiv:0903.2514 [math.NT] (2009) constant Q_1^(1).
G. Niklasch, Some number theoretical constants: 1000-digit values.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.
R. Sitaramachandrarao and D. Suryanarayana, On Sigma_{n<=x} sigma*(n) and Sigma_{n<=x} phi*(n), Proceedings of the American Mathematical Society, Vol. 41, No. 1 (1973), pp. 61-66.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1,arXiv preprint, arXiv:1608.00795 [math.NT], 2016.
Deyu Zhang and Wenguang Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq., Vol. 13 (2010), Article 10.4.7, eq. (4).
Rimer Zurita Generalized Alternating Sums of Multiplicative Arithmetic Functions, J. Int. Seq., Vol. 23 (2020), Article 20.10.4.

Cf. A001615, A007947, A047994, A078082, A268335, A306633.

Cf. A000203, A065473, A065480, A065490, A008683.

$MaxExtraPrecision = 1200; digits = 98; terms = 1200; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

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

From Amiram Eldar, Mar 05 2019: (Start)
Equals lim_{m->oo} (2/m^2)*Sum_{k=1..m} rad(k), where rad(k) = A007947(k) is the squarefree kernel of k (Cohen).
Equals lim_{m->oo} (2/m^2)*Sum_{k=1..m} uphi(k), where uphi(k) = A047994(k) is the unitary totient function (Sitaramachandrarao and Suryanarayana).
Equals lim_{m->oo} (1/log(m))*Sum_{k=1..m} 1/psi(k), where psi(k) = A001615(k) is the Dedekind psi function (Sita Ramaiah and Suryanarayana).
(End)
Equals A065473*A013661/A065480. - Luke Palmer, Apr 27 2019
Equals Sum_{k>=1} mu(k)/(k*sigma(k)), where mu is the Möbius function (A008683) and sigma(k) is the sum of divisors of k (A000203). - Amiram Eldar, Jan 14 2022
Equals 1/A065489. - R. J. Mathar, May 27 2025

A065464 Decimal expansion of Product_{p prime} (1 - (2*p-1)/p^3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

Sum_{n <= x} A189021(n) ~ kx, where k is this constant. - Charles R Greathouse IV, Jan 24 2018

The probability that a number chosen at random is squarefree and coprime to another randomly chosen random (see Schroeder, 2009). - Amiram Eldar, May 23 2020, corrected Aug 04 2020

			0.428249505677094440218765707581823546...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.5.1, p. 110.
Manfred Schroeder, Number Theory in Science and Communication, 5th edition, Springer, 2009, page 59.

R. J. Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011; Equation (116).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A000010, A057434, A078078, A056671.
Cf. A065463, A013661.
Cf. A065473, A065480.

$MaxExtraPrecision = 800; digits = 98; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms+10]]; r[n_Integer] := LR[[n]]; (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n-1]/(n-1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

prodeulerrat(1 - (2*p-1)/p^3) \\ Amiram Eldar, Mar 12 2021

Equals A065463 divided by A013661. - R. J. Mathar, Mar 22 2011
Equals A065473 divided by A065480. - R. J. Mathar, May 02 2019
Equals (6/Pi^2)^2 * Product_{p prime} (1 + 1/(p^3 + p^2 - p - 1)) = 1.1587609... * (6/Pi^2)^2. - Amiram Eldar, May 23 2020
Equals lim_{m->oo} (1/m) * Sum_{k==1..m} (phi(k)/k)^2, where phi is the Euler totient function (A000010). - Amiram Eldar, Mar 12 2021

More digits from Vaclav Kotesovec, Dec 18 2019

A088453 Decimal expansion of 1/zeta(3).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Sep 30 2003

This is the probability that three randomly chosen integers are relatively prime (see A018805). - Gary McGuire, Dec 13 2004
This is also the probability that a random integer is cubefree. - Eugene Salamin, Dec 13 2004
On the other hand, the probability that three randomly-chosen integers are pairwise relatively prime is given by A065473. - Charles R Greathouse IV, Nov 14 2011
This is also the 'probability' that a random algebraic number's denominator is equal to its leading coefficient, see Arno, Robinson, & Wheeler. - Charles R Greathouse IV, Nov 12 2014
This is the probability that a random point on a cubic lattice is visible from the origin, i.e., there is no other lattice point that lies on the line segment between this point and the origin. - Amiram Eldar, Jul 08 2020

			0.831907372580707468683126278821530734417...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6, p. 41.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.

Steven Arno, M. L. Robinson, and Ferell S. Wheeler, On denominators of algebraic numbers and integer polynomials, Journal of Number Theory 57:2 (April 1996), pp. 292-302.
Eric Weisstein's World of Mathematics, Relatively Prime.

Cf. A002117, A008683.

RealDigits[1/Zeta[3],10,120][[1]] (* Harvey P. Dale, May 31 2019 *)

fpprec : 200$ bfloat( 1/zeta(3))$ bfloat(%); /* Martin Ettl, Oct 15 2012 */

1/zeta(3) \\ Charles R Greathouse IV, Nov 12 2014

Equals 1/A002117.
From Amiram Eldar, Aug 20 2020: (Start)
Equals Sum_{k>=1} mu(k)/k^3, where mu is the Möbius function (A008683).
Equals Product_{p prime} (1 - 1/p^3). (End)

Entry revised by N. J. A. Sloane, Dec 16 2004

A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 19 2019

The asymptotic density of A337050. - Amiram Eldar, Aug 13 2020

			0.748535259682363564644215048637910601641640343005324404515852793925925...

Cf. A057723, A065464, A065473, A065476, A065487, A328017, A330594, A330595, A337050, A372636.

Do[Print[N[Exp[-Sum[q = Expand[(p^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 + 1/p^3) \\ Amiram Eldar, Mar 17 2021

Equals (6/Pi^2) * A065487. - Amiram Eldar, Jun 10 2020

A000189 Number of solutions to x^3 == 0 (mod n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 16, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1, 4, 1, 3

Offset: 1

N. J. A. Sloane

Shadow transform of the cubes A000578. - Michel Marcus, Jun 06 2013

			a(4) = 2 because 0^3 == 0, 1^3 == 1, 2^3 == 0, and 3^3 == 3 (mod 4); also, a(9) = 3 because 0^3 = 0, 3^3 == 0, and 6^3 = 0 (mod 9), while x^3 =/= 0 (mod 9) for x = 1, 2, 4, 5, 7, 8. - _Petros Hadjicostas_, Sep 16 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150. (ps, pdf); see Definition 7 for the shadow transform.
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).
OEIS Wiki, Shadow transform.
N. J. A. Sloane, Transforms.

Cf. A000578, A019555.

Array[ Function[ n, Count[ Array[ PowerMod[ #, 3, n ]&, n, 0 ], 0 ] ], 100 ]
f[p_, e_] := p^Floor[2*e/3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n)=my(f=factor(n));prod(i=1,#f[,1],f[i,1]^(2*f[i,2]\3)) \\ Charles R Greathouse IV, Jun 06 2013

for(n=1, 100, print1(direuler(p=2, n, (1 + X + p*X^2)/(1 - p^2*X^3))[n], ", ")) \\ Vaclav Kotesovec, Aug 30 2021

Multiplicative with a(p^e) = p^[2e/3]. - David W. Wilson, Aug 01 2001
a(n) = n/A019555(n). - Petros Hadjicostas, Sep 15 2019
Dirichlet g.f.: zeta(3*s-2) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1)). - Amiram Eldar, Sep 09 2023
From Vaclav Kotesovec, Sep 09 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * zeta(3*s-2) * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s-1) - 1/p^(4*s-2) + 1/p^(4*s-1) + 1/p^(5*s-2)).
Let f(s) = Product_{primes p} (1 - 1/p^(2*s) - 1/p^(3*s-1) - 1/p^(4*s-2) + 1/p^(4*s-1) + 1/p^(5*s-2)).
Sum_{k=1..n} a(k) ~ (f(1)*n/6) * (log(n)^2/2 + (6*gamma - 1 + f'(1)/f(1))*log(n) + 1 - 6*gamma + 11*gamma^2 - 14*sg1 + (6*gamma - 1)*f'(1)/f(1) + f''(1)/(2*f(1))), where
f(1) = Product_{primes p} (1 - 3/p^2 + 2/p^3) = A065473 = 0.2867474284344787341078927127898384464343318440970569956414778593366522431...,
f'(1) = f(1) * Sum_{primes p} 9*log(p) / (p^2 + p - 2) = f(1) * 4.1970213428422788650375569145777616746065054412058004220013841318980729375...,
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} (-29*p^2 - 17*p + 1) * log(p)^2 / (p^2 + p - 2)^2 = f'(1)^2/f(1) + f(1) * (-21.3646716550082193262514333696570765444176783899223644201265894338042468...),
gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

A055076 Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 4, 1, 4, 2, 2, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 4, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 4, 4, 4, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4, 4, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 2, 1, 2, 8, 2, 4, 8

Offset: 1

Labos Elemer, Jun 13 2000

Number of distinct values of gcd(d, n!/d) if d runs over divisors of n! seems to be A046951(n).
a(n) = 1 iff n is a square. - Bernard Schott, Oct 22 2019
a(n) is the number of the unitary divisors (cf. A077610) of n that are exponentially odd (A268335). - Amiram Eldar, Nov 11 2022
The number of infinitary divisors of n that are squarefree (A005117). - Amiram Eldar, Jan 09 2024

			n=120, the set of gcd(d, 120/d) values for the 16 divisors of 120 is {1,2,1,2,1,2,1,2,2,1,2,1,2,1,2,1}. The max is 2 and it occurs 8 times, so a(120)=8. This sequence seems to consist of powers of 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A005117, A007913, A034444, A077610, A162642, A268335, A295316, A350389.

with(numtheory):
a:= n->(p->coeff(p, x, degree(p)))(add(x^igcd(d, n/d), d=divisors(n))):
seq(a(n), n=1..105);  # Alois P. Heinz, Jul 21 2015

a[n_] := With[{g = GCD[#, n/#]& /@ Divisors[n]}, Count[g, Max[g]]];
Array[a, 105] (* Jean-François Alcover, Mar 28 2017 *)
f[p_, e_] := 2^Mod[e, 2]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Nov 11 2022 *)

A055076(n) = if(1==n,n,my(es=factor(n)[,2]~); prod(i=1,#es,2^(es[i]%2))); \\ Antti Karttunen, Apr 05 2021

;; With memoization-macro definec.
(definec (A055076 n) (if (= 1 n) n (* (+ 1 (A000035 (A067029 n))) (A055076 (A028234 n))))) ;; Antti Karttunen, Dec 02 2017

Multiplicative with a(p^e) = 2^(e mod 2). - Vladeta Jovovic, Dec 13 2002
a(n) = 2^A162642(n). - Antti Karttunen, Dec 02 2017
a(n) = A034444(A007913(n)). [Found by LODA miner, see C. Krause link. Essentially the same formula as the above ones] - Antti Karttunen, Apr 05 2021
From Amiram Eldar, Sep 09 2023: (Start)
a(n) = A034444(A350389(n)).
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 2/p^s). (End)
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - 3/p^(2*s) + 2/p^(3*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * f(s).
Sum_{k=1..n} a(k) ~ (Pi^2 * f(1) * n / 6) * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.286747428434478734107892712789838446434331844097056995641477859336652243...,
f'(1) = f(1) * Sum_{primes p} 6*log(p) / (p^2 + p - 2) = f(1) * 2.798014228561519243358371276385174449737670294137200281334256087932048625...
and gamma is the Euler-Mascheroni constant A001620. (End)

A330595 Decimal expansion of Product_{primes p} (1 + 1/p^2 + 1/p^3).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Dec 19 2019

			1.748932997843245303033906997685114802259883493595480897273662144084849...

Cf. A065464, A065473, A065476, A160889, A328017, A330594, A330596, A338325.

Do[Print[N[Exp[-Sum[q = Expand[(-p^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 + 1/p^3) \\ Vaclav Kotesovec, Sep 19 2020

Equals Sum_{n>=1} 1/A338325(n). - Amiram Eldar, Oct 26 2020

A256392 Decimal expansion of Product_{p prime} (1-4/p^2+4/p^3-1/p^4).

Original entry on oeis.org

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

Offset: 0

Juan Arias-de-Reyna, Mar 28 2015

Also decimal expansion of the probability that an integer tuple (x,y,z,w) satisfies gcd(x,y) = gcd(y,z) = gcd(z,w) = gcd(w,x) = 1.

			0.2177787166195363783230075141...

Juan Arias-de-Reyna, Table of n, a(n) for n = 0..1000
Juan Arias de Reyna and Randell Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq., Vol. 18 (2015), Article 15.10.4; arXiv preprint, arXiv:1403.2769 [math.NT], 2014.

Cf. A065473, A126775, A196524, A256391, A330523, A336643.

Do[Print[N[Exp[-Sum[q = Expand[(4 p^2 - 4 p^3 + p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 50]], {t, 10, 100, 10}] (* Vaclav Kotesovec, Dec 17 2019 *)

prodeulerrat(1-4/p^2+4/p^3-1/p^4) \\ Amiram Eldar, Mar 03 2021

A072048 Number of divisors of the squarefree numbers: tau(A005117(n)).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 8, 2, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 8, 2, 4, 8, 2, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 2, 2, 8, 2, 8, 4, 2, 2, 8, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4, 8, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 2, 8, 4, 2

Offset: 1

Reinhard Zumkeller, Jun 09 2002

nonn

Also the number of cubefree numbers with the same squarefree kernel as the n-th squarefree number, see A073245.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
B. Gordon and K. Rogers, Sums of the divisor function, Canadian Journal of Mathematics, Vol. 16 (1964), pp. 151-158.

Cf. A000005, A001620, A062822, A065473.

Cf. A000079, A001221, A005117, A072047.

a072048 = (2 ^) . a072047  -- Reinhard Zumkeller, Dec 13 2015

A072048:=n->`if`(numtheory[issqrfree](n) = true, numtheory[tau](n), NULL); seq(A072048(k), k=1..100); # Wesley Ivan Hurt, Oct 13 2013

DivisorSigma[0, Select[Range[200], SquareFreeQ]] (* Amiram Eldar, Oct 29 2022 *)

from math import isqrt
from sympy import mobius, divisor_count
def A072048(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return divisor_count(m) # Chai Wah Wu, Aug 12 2024

a(n) = A000005(A005117(n)).
a(n) = 2^A072047(n) = 2^A001221(A005117(n)).
Sum_{k=1..n} a(k) ~ A * n * log(n) + B * n + O(n^(1/2+eps)), where  A = A065473, B = A * ((2*gamma-1) + 6 * Sum_{p prime} (p-1)*log(p)/(p^2*(p+2)) = 0.236184..., and gamma = A001620 (Gordon and Rogers, 1964). - Amiram Eldar, Oct 29 2022

cp's OEIS Frontend

A078073 Continued fraction for constant defined in A065473.

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A065463 Decimal expansion of Product_{p prime} (1 - 1/(p*(p+1))).

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A065464 Decimal expansion of Product_{p prime} (1 - (2*p-1)/p^3).

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A088453 Decimal expansion of 1/zeta(3).

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A330596 Decimal expansion of Product_{primes p} (1 - 1/p^2 + 1/p^3).

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A000189 Number of solutions to x^3 == 0 (mod n).

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A055076 Multiplicity of Max{gcd(d, n/d)} when d runs over divisors of n.

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