A065472 -id:A065472 - cp's OEIS frontend

A078091 Continued fraction for constant defined in A065472.

0, 1, 3, 2, 6, 13, 1, 2, 2, 3, 1, 1, 6, 1, 4, 1, 10, 1, 5, 1, 13, 1, 12, 384, 1, 2, 1, 14, 1, 11, 1, 2, 33, 1, 4, 1, 8, 1, 1, 1, 1, 2, 2, 1, 1, 11, 1, 1, 1, 8, 7, 1, 1, 5, 6, 7, 1, 1, 2, 3, 1, 8, 1, 1, 1, 2, 1, 1, 1, 7, 1, 2, 1, 2, 1, 15, 1, 2, 2, 15, 1, 2, 2, 1, 1, 8, 3, 2, 1, 1, 3, 13, 1

Offset: 0

Benoit Cloitre, Dec 02 2002

Cf. A065472 (decimal expansion).

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

Offset changed by Andrew Howroyd, Jul 05 2024

A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

Also decimal expansion of the probability that an integer triple (x, y, z) is pairwise coprime. - Charles R Greathouse IV, Nov 14 2011

The probability that 2 numbers chosen at random are coprime, and both squarefree (Delange, 1969). - Amiram Eldar, Aug 04 2020

			0.2867474284344787341078927127898384...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6, p. 41.
Gerald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd edition, American Mathematical Society, 2015, page 59, exercise 55 and 56.

Juan Arias de Reyna, R. Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq. 18 (2015) 15.10.4
Tim Browning, The divisor problem for binary cubic forms, Journal de théorie des nombres de Bordeaux, Vol. 23, No. 3 (2011), pp. 579-602; arXiv preprint, arXiv:1006.3476 [math.NT], 2010.
Hubert Delange, On some sets of pairs of positive integers, Journal of Number Theory, Vol. 1, No. 3 (1969), pp. 261-279. See p. 277.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 181.
G. Niklasch, Some number theoretical constants: 1000-digit values.
G. Niklasch, Some number theoretical constants: 1000-digit values. [cached copy]
László Tóth, The probability that k positive integers are pairwise relatively prime, Fibonacci Quart., Vol. 40 (2002), pp. 13-18.
László Tóth, Another generalization of Euler's arithmetic function and of Menon's identity, arXiv:2006.12438 [math.NT], 2020. See p. 3.
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A065472, A069201, A069212, A074816, A074823, A078073, A098198, A227929, A299822.

digits = 100; NSum[-(2+(-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 2 digits, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 11 2016 *)

prodeulerrat(1 - (3*p-2)/(p^3)) \\ Amiram Eldar, Mar 17 2021

Equals Prod_{p prime} (1 - 1/p)^2*(1 + 2/p). - Michel Marcus, Apr 16 2016
The constant c in Sum_{k<=x} mu(k)^2 * 2^omega(k) = c * x * log(x) + O(x), where mu is A008683 and omega is A001221, and in Sum_{k<=x} 3^omega(k) = (1/2) * c * x * log(x)^2 + O(x*log(x)) (see Tenenbaum, 2015). - Amiram Eldar, May 24 2020
Equals A065472 * A227929 =  A065472 / A098198. - Amiram Eldar, Aug 04 2020

Name corrected by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 03 2003

More digits from Vaclav Kotesovec, Dec 19 2019

A307868 Decimal expansion of the asymptotic mean of phi(k)/psi(k), where phi(k) is Euler totient function (A000010) and psi(k) is Dedekind psi function (A001615).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 02 2019

Also, the asymptotic mean of A162511. - Amiram Eldar, Sep 18 2022

			0.47168061361299786807523563308048208742592638200698...

V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link.

Cf. A000010, A001615, A013661, A065472, A071974, A162511, A173557.

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-2, 1, 2}, {0, -4, 6}, m]; RealDigits[(2/3) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

prodeulerrat(1 - 2/(p*(p+1))) \\ Vaclav Kotesovec, Sep 19 2020

Equals lim_{m->oo} (1/m)*Sum_{k=1..m} phi(k)/psi(k).
Equals Product_{p prime} (1 - 2/(p * (p+1))).
Equals A065472 / zeta(2). - Amiram Eldar, Sep 18 2022

More digits from Vaclav Kotesovec, Sep 19 2020

A038063 Product_{k>=1}1/(1 - x^k)^a(k) = 1 + 2x.

Original entry on oeis.org

2, -3, 2, -3, 6, -11, 18, -30, 56, -105, 186, -335, 630, -1179, 2182, -4080, 7710, -14588, 27594, -52377, 99858, -190743, 364722, -698870, 1342176, -2581425, 4971008, -9586395, 18512790, -35792449, 69273666, -134215680, 260300986

Offset: 1

Christian G. Bower, Jan 04 1999

sign

Apart from initial terms, exponents in expansion of A065472 as a product zeta(n)^(-a(n)).

Seiichi Manyama, Table of n, a(n) for n = 1..3000
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
N. J. A. Sloane, Euler transform.

Cf. A001037, A008683, A038064, A038065, A038066, A038067, A038068, A038069, A038070, A059966, A060477, A065472.

a[n_] := DivisorSum[n, (-1)^(#+1) * MoebiusMu[n/#]*2^# &] / n; Array[a, 33] (* Amiram Eldar, May 29 2025 *)

{a(n)=polcoeff(sum(k=1,n,moebius(k)/k*log(1+2*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Oct 13 2010

a(n) = (1/n) * Sum_{d divides n} (-1)^(d+1)*moebius(n/d)*2^d. - Vladeta Jovovic, Sep 06 2002
G.f.: Sum_{n>=1} moebius(n)*log(1 + 2*x^n)/n, where moebius(n) = A008683(n). - Paul D. Hanna, Oct 13 2010
For n == 0, 1, 3 (mod 4), a(n) = (-1)^(n+1)*A001037(n), which for n>1 also equals (-1)^(n+1)*A059966(n) = (-1)^(n+1)*A060477(n).
For n == 2 (mod 4), a(n) = -(A001037(n) + A001037(n/2)). - George Beck and Max Alekseyev, May 23 2016
a(n) ~ -(-1)^n * 2^n / n. - Vaclav Kotesovec, Jun 12 2018

A162644 Numbers m such that A162511(m) = +1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97, 100

Offset: 1

Reinhard Zumkeller, Jul 08 2009

nonn

Also numbers n with A008836(n)=(-1)^A001221(n). - Enrique Pérez Herrero, Aug 03 2012

This sequence has an asymptotic density (1 + A065472/zeta(2))/2 = 0.735840... (Mossinghoff and Trudgian, 2019). - Amiram Eldar, Jul 07 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.

Complement of A162645.
A002035 is a subsequence.
Cf. A008836, A001221, A036537, A065472.

Select[Range[100], EvenQ[PrimeOmega[#] - PrimeNu[#]] &] (* Amiram Eldar, Jul 07 2020 *)

A162645 Numbers m such that A162511(m) = -1.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 98, 99, 108, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228

Offset: 1

Reinhard Zumkeller, Jul 08 2009

nonn

Numbers n where A001222(n)-A001221(n) is odd. - Enrique Pérez Herrero, Jul 07 2012

This sequence has an asymptotic density (1 - A065472/zeta(2))/2 = 0.264159... (Mossinghoff and Trudgian, 2019). - Amiram Eldar, Jul 07 2020

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.

Complement of A162644.
Subsequence of A072587.
Cf. A001221, A001222, A065472, A162643, A162511.

Select[Range[10^2],OddQ[PrimeOmega[#]-PrimeNu[#]]&] (* Enrique Pérez Herrero, Jul 07 2012 *)

A110833 a(n) = (prime(n)+1)^2.

Original entry on oeis.org

9, 16, 36, 64, 144, 196, 324, 400, 576, 900, 1024, 1444, 1764, 1936, 2304, 2916, 3600, 3844, 4624, 5184, 5476, 6400, 7056, 8100, 9604, 10404, 10816, 11664, 12100, 12996, 16384, 17424, 19044, 19600, 22500, 23104, 24964, 26896, 28224, 30276, 32400, 33124, 36864

Offset: 1

Giovanni Teofilatto, Sep 18 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A008864, A065472, A065486.

[(p+1)^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 27 2014

Table[(Prime[n] + 1)^2, {n, 200}] (* Vincenzo Librandi, Mar 27 2014 *)

from sympy import primerange
print([(p+1)**2 for p in primerange(1, 192)]) # Michael S. Branicky, Sep 16 2021

From Amiram Eldar, Jan 23 2021: (Start)
a(n) = A008864(n)^2.
Product_{n>=1} (1 + 1/a(n)) = A065486.
Product_{n>=1} (1 - 1/a(n)) = A065472. (End)
Sum 1/a(n) = A382554. - R. J. Mathar, Mar 31 2025

Corrected and extended by Ray Chandler, Oct 08 2005

A116393 Decimal expansion of Product(1 - 1/(p+1)^3), p prime >= 2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Apr 15 2006

Calculated from A065465 * A065467^3 * X / A065466^2 where X = 1.03108637675008536... = product[ 1+ (4p^13 +13p^12 +10p^11 -3p^10 -9p^9 -12p^8 -9p^7 +6p^5 +4p^4 +2p^3 +p^2 -p -1) /(p^5+p^4-1)^3 / (p^3+p^2-1) / (p+1)] over prime p >=2. - R. J. Mathar, Sep 10 2007

			0.9403004...

Cf. A065472.

prodeulerrat(1 - 1/(p+1)^3) \\ Amiram Eldar, Nov 29 2020

More terms from John W. Layman and Zak Seidov, Apr 20 2006
More terms from R. J. Mathar, Sep 10 2007
More terms from Amiram Eldar, Nov 29 2020

A340820 Decimal expansion of (1 + Product_{p prime} (1 - 2/(p*(p+1))))/2.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jan 22 2021

The asymptotic density of numbers k such that A046660(k) is even (A162644).

Detrey et al. (2016) calculated 1000 decimal digits of this constant.

			0.735840306806498934037617816540241043712963191003493...

Jérémie Detrey, Pierre-Jean Spaenlehauer and Paul Zimmermann, Computing the rho constant, 2016.
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019. See page 9.

Cf. A013661, A046660, A065472, A162644, A340818, A340819.

PARI
```
(prodeulerrat(1 - 2/(p*(p+1))) + 1)/2
```

Equals (1 + A065472/zeta(2))/2.

Equals lim_{n->oo} A340818(n)/A340819(n).

cp's OEIS Frontend

A078091 Continued fraction for constant defined in A065472.

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A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

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A307868 Decimal expansion of the asymptotic mean of phi(k)/psi(k), where phi(k) is Euler totient function (A000010) and psi(k) is Dedekind psi function (A001615).

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A038063 Product_{k>=1}1/(1 - x^k)^a(k) = 1 + 2x.

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A162644 Numbers m such that A162511(m) = +1.

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A162645 Numbers m such that A162511(m) = -1.

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A110833 a(n) = (prime(n)+1)^2.

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A116393 Decimal expansion of Product(1 - 1/(p+1)^3), p prime >= 2.

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