cp's OEIS Frontend

A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

%I A327837 #18 Dec 24 2024 07:30:48
%S A327837 1,6,0,2,3,1,7,1,0,2,3,0,5,4,1,8,0,5,2,3,4,9,6,2,6,3,1,5,6,2,1,1,6,1,
%T A327837 0,0,3,7,7,6,9,3,9,4,9,5,7,8,5,5,7,2,7,3,7,7,4,6,5,3,5,2,8,5,9,8,7,8,
%U A327837 8,8,8,6,0,2,1,6,3,3,5,4,7,2,7,5,6,6,7,3,3,9,0,4,9,4,8,8,0,6,4,1,8,0,7,5,7
%N A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).
%H A327837 Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant Z3).
%H A327837 V. Sita Ramaiah and D. Suryanarayana, <a href="https://web.archive.org/web/20200803214209/http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005bab_1334.pdf">Sums of reciprocals of some multiplicative functions - II</a>, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355 (eq. 2.37 and 3.18, pp. 1346 and 1354).
%H A327837 Abdelhakim Smati and Jie Wu, <a href="http://elib.mi.sanu.ac.rs/files/journals/publ/81/n075p021.pdf">On the exponential divisor function</a>, Publications de l'Institut Mathématique, Vol. 61 (1997), pp. 21-32.
%H A327837 László Tóth, <a href="http://publi.math.unideb.hu/load_jpg.php?p=1257">An order result for the exponential divisor function</a>, Publ. Math. Debrecen, Vol. 71, No. 1-2 (2007), pp. 165-171, <a href="https://arxiv.org/abs/0708.3552">arXiv preprint,</a>, arXiv:0708.3552 [math.NT], 2007.
%H A327837 László Tóth, <a href="http://emis.ams.org/journals/JIS/VOL20/Toth/toth25.html">Alternating sums concerning multiplicative arithmetic functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.10, p. 30).
%H A327837 Jie Wu, <a href="http://jtnb.cedram.org/item?id=JTNB_1995__7_1_133_0">Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré</a>, Journal de théorie des nombres de Bordeaux, Vol. 7, No. 1, (1995), pp. 133-141.
%F A327837 Equals lim_{k->oo} A145353(k)/k.
%F A327837 Equals Product_{p prime} (1 + Sum_{e >= 2} p^(-e) * (d(e) - d(e-1))), where d(e) is the number of divisors of e (A000005).
%F A327837 Equals Product_{p prime} (1 - 1/p) * (2 - (log(p-1) + QPolyGamma(0, 1, 1/p)) / log(p)). - _Vaclav Kotesovec_, Feb 27 2023
%F A327837 From _Amiram Eldar_, Dec 24 2024: (Start)
%F A327837 Equals lim_{m->oo} (1/m) * Sum_{k=1..m} k/uphi(k) = lim_{m->oo} (1/m) * Sum_{k=1..m} A319677(k)/A319676(k), where uphi(k) is the unitary totient function (A047994).
%F A327837 Equals lim_{m->oo} (1/log(m)) * Sum_{k=1..m} 1/uphi(k) = lim_{m->oo} (1/log(m)) * A379517(m)/A379518(m).
%F A327837 Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A361967(k).
%F A327837 Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k-1))).
%F A327837 Equals Product_{p prime} (1 + (1-1/p) * Sum_{k>=1} 1/(p^k*(p^k-1))). (End)
%e A327837 1.602317102305418052349626315621161003776939495785572...
%t A327837 $MaxExtraPrecision = 1500; m = 1500; em = 500; f[x_] := 1 + Log[1 + Sum[x^e * (DivisorSigma[0, e] - DivisorSigma[0, e - 1]), {e, 2, em}]]; c = Rest[ CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m] ]; RealDigits[ Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
%Y A327837 Cf. A000005, A047994, A049419, A145353, A361013, A361967, A379517, A379518.
%Y A327837 Cf. A059956 (constant for unitary divisors), A306071 (bi-unitary), A327576 (infinitary).
%K A327837 nonn,cons
%O A327837 1,2
%A A327837 _Amiram Eldar_, Sep 27 2019
%E A327837 More digits from _Vaclav Kotesovec_, Jun 13 2021