A358659 Decimal expansion of the asymptotic mean of the ratio between the number of exponential unitary divisors and the number of exponential divisors.

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

Offset: 0

Author

Amiram Eldar, Nov 25 2022

Keywords

• nonn
• cons

Examples

			0.984883641877228294095370138048961137647316322227058...
		

Links

• Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp. Vol. 35 (2011), pp. 205-216.

Crossrefs

Cf. A049419, A278908.

Similar sequences: A307869, A308042, A308043.

Programs

Mathematica

r[n_] := 2^PrimeNu[n]/DivisorSigma[0, n]; $MaxExtraPrecision = 500; m = 500; f[x_] := Log[1 + Sum[x^e*(r[e] - r[e - 1]), {e, 4, m}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[f[1/2] + NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

Formula

Equals lim_{m->oo} (1/m) Sum_{k=1..m} A278908(k)/A049419(k).

Equals Product_{p prime} (1 + Sum_{e >= 4} (r(e) - r(e-1))/p^e), where r(e) = A278908(e)/A049419(e).