A306071 Decimal expansion of Sum_{n>=1} (-1)^omega(n) phi(n)^2/n^4, where omega(n) is the number of distinct prime factors of n (A001221) and phi is Euler's totient function (A000010).

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

Offset: 0

Author

Amiram Eldar, Jun 19 2018

Keywords

• nonn
• cons

Comments

The constant A that appears in the asymptotic formulae for the sums of the bi-unitary divisor function (A306069) and the bi-unitary totient function (A306070).

The product in Suryanarayana's 1972 paper has a error that was corrected in his 1975 paper.

The probability that 2 randomly selected numbers will be unitary coprime (i.e. their largest common unitary divisor is 1). - Amiram Eldar, Aug 27 2019

Examples

			0.80733082163620503914...
		

References

• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 72.

Links

• D. Suryanarayana, The number of bi-unitary divisors of an integer, The theory of arithmetic functions, ed. Anthony A. Gioia and Donald L. Goldsmith, Springer, Berlin, Heidelberg, 1972, pp. 273-282.
• D. Suryanarayana and R. Sita Rama Chandra Rao, The number of bi-unitary divisors of an integer - II, Journal of the Indian mathematical Society, Vol. 39, No. 1-4 (1975), pp. 261-280.
• László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.2.
• 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. 508), preprint, arXiv:1310.7053 [math.NT] (2014) (see p. 21).

Crossrefs

Cf. A000010, A001221, A286324, A306069, A306070.

Programs

Mathematica

cc = CoefficientList[Series[Log[1 - (p - 1)/(p^2*(p + 1))] /. p -> 1/x, {x, 0, 36}], x]; f = FindSequenceFunction[cc]; digits = 20; A = Exp[NSum[f[n + 1 // Floor]*(PrimeZetaP[n]), {n, 2, Infinity}, NSumTerms -> 16 digits, WorkingPrecision -> 16 digits]]; RealDigits[A, 10, digits][[1]] (* Jean-François Alcover, Jun 19 2018 *)
$MaxExtraPrecision = 1000; Do[Print[Zeta[2] * Exp[-N[Sum[q = Expand[(2*p^2 - 2*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}], 120]]], {t, 300, 1000, 100}] (* Vaclav Kotesovec, May 29 2020 *)

PARI

prodeulerrat(1 - (p-1)/(p^2 * (p+1))) \\ Amiram Eldar, Mar 18 2021

Formula

Equals Product_{p prime} (1 - (p-1)/(p^2 * (p+1))).

Equals zeta(2) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 1/p^4).

Extensions

a(1)-a(20) from Jean-François Alcover, Jun 19 2018

a(20)-a(24) from Jon E. Schoenfield, May 27 2019

More terms from Vaclav Kotesovec, May 29 2020