OFFSET
0,1
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
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).
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).
EXAMPLE
0.80733082163620503914...
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 *)
PROG
(PARI) prodeulerrat(1 - (p-1)/(p^2 * (p+1))) \\ Amiram Eldar, Mar 18 2021
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jun 19 2018
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
STATUS
approved