A358658 Decimal expansion of the asymptotic mean of the e-unitary Euler function (A321167).

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

Offset: 1

Author

Amiram Eldar, Nov 25 2022

Keywords

• nonn
• cons

Examples

			1.307321371706072369296422804253988391427434686082394...
		

Links

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

Crossrefs

Cf. A047994, A321167, A327838.

Programs

Mathematica

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; $MaxExtraPrecision = 500; m = 500; fun[x_] := Log[1 + Sum[x^e*(uphi[e] - uphi[e - 1]), {e, 3, m}]]; c = Rest[CoefficientList[Series[fun[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[fun[1/2] + NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Formula

Equals lim_{m->oo} (1/m) Sum_{k=1..m} A321167(k).

Equals Product_{p prime} (1 + Sum_{e >= 3} (uphi(e) - uphi(e-1))/p^e), where uphi is the unitary totient function (A047994).