A307160 Decimal expansion of the constant c in the asymptotic formula for the partial sums of the bi-unitary divisors sum function, A307159(k) ~ c*k^2.

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

Offset: 0

Author

Amiram Eldar, Mar 27 2019

Keywords

• nonn
• cons

Comments

The asymptotic mean of the bi-unitary abundancy index lim_{n->oo} (1/n) * Sum_{k=1..n} A188999(k)/k = 2*c = 1.505677... - Amiram Eldar, Jun 10 2020

Examples

			0.75283874100229431154333095155304127651952546756522...
		

References

• D. Suryanarayana and M. V. Subbarao, Arithmetical functions associated with the biunitary k-ary divisors of an integer, Indian J. Math., Vol. 22 (1980), pp. 281-298.

Links

• László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1, section 4.13.

Crossrefs

Cf. A188999, A306071, A306072, A307159.

Programs

Mathematica

$MaxExtraPrecision = 1000; nm=1000; c = Rest[CoefficientList[Series[Log[1 - 2*x^3 + x^4 + x^5 - x^6],{x,0,nm}],x] * Range[0, nm]]; RealDigits[(Zeta[2]*Zeta[3]/2) * Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]

Formula

Equals (zeta(2)*zeta(3)/2)* Product_{p}(1 - 2/p^3 + 1/p^4 + 1/p^5 - 1/p^6).

Extensions

More terms from Vaclav Kotesovec, May 29 2020