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%I #18 Jun 19 2023 12:04:54
%S 1,5,30,198,1456,11751,97385,829157,7266286,64782731,582798892
%N The number of practical numbers not exceeding 10^n.
%H Maurice Margenstern, <a href="http://dx.doi.org/10.1016/S0022-314X(05)80022-8">Les nombres pratiques: théorie, observations et conjectures</a>, Journal of Number Theory 37 (1): 1-36, 1991.
%H Andreas Weingartner, <a href="https://doi.org/10.1093/qmath/hav006">Practical numbers and the distribution of divisors</a>, Q. J. Math. 66 (2015), 743 - 758.
%H Andreas Weingartner, <a href="https://arxiv.org/abs/1705.06349">On the constant factor in several related asymptotic estimates</a>, arXiv preprint arXiv:1705.06349 [math.NT], 2017-2018.
%H Andreas Weingartner, <a href="https://arxiv.org/abs/1906.07819">The constant factor in the asymptotic for practical numbers</a>, arXiv:1906.07819 [math.NT], 2019.
%F a(n) ~ c * f(10^n), where f(x) = x/log(x) and c is a constant (evaluated as 1.341 by Margenstern; Weingartner proved that 1.311 < c < 1.693).
%F 1.33606 < c < 1.33609. See Weingartner (2019). - _Michel Marcus_, Jun 19 2019
%t practicalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; n=0; s={}; Do[If[k>10^n, AppendTo[s,c]; n++]; If[practicalQ [k],c++], {k,1,100000}]; s (* after _T. D. Noe_ at A005153 *)
%o (PARI) my(x=1, i=0); for(k=1, oo, if(is_A005153(k), i++); if(k >= x, print1(i, ", "); x=x*10)) \\ _Felix Fröhlich_, Dec 08 2018. [Stale copy of is_A005153 removed here. Please do not duplicate code, it will necessarily become obsolete or worse. - _M. F. Hasler_, Jun 19 2023]
%Y Cf. A005153, A209237, A273773.
%K nonn,more
%O 0,2
%A _Amiram Eldar_, Dec 01 2018
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