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

0,1

Links

Table of n, a(n) for n=0..104.

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

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).

Example

0.984883641877228294095370138048961137647316322227058...

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]]

Crossrefs

Cf. A049419, A278908.

Similar sequences: A307869, A308042, A308043.

Sequence in context: A200117 A019889 A243266 * A010548 A011458 A343057

Adjacent sequences: A358656 A358657 A358658 * A358660 A358661 A358662

Keyword

nonn,cons

Author

Amiram Eldar, Nov 25 2022

Status

approved