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A358658
Decimal expansion of the asymptotic mean of the e-unitary Euler function (A321167).
1
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
1,2
LINKS
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} 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).
EXAMPLE
1.307321371706072369296422804253988391427434686082394...
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]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Nov 25 2022
STATUS
approved