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A322257
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The number of practical numbers not exceeding 10^n.
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3
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1, 5, 30, 198, 1456, 11751, 97385, 829157, 7266286, 64782731, 582798892
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OFFSET
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0,2
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LINKS
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FORMULA
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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).
1.33606 < c < 1.33609. See Weingartner (2019). - Michel Marcus, Jun 19 2019
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MATHEMATICA
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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 *)
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PROG
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(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]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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