A322257 The number of practical numbers not exceeding 10^n.

1, 5, 30, 198, 1456, 11751, 97385, 829157, 7266286, 64782731, 582798892

Offset

0,2

Links

Table of n, a(n) for n=0..10.

Maurice Margenstern, Les nombres pratiques: théorie, observations et conjectures, Journal of Number Theory 37 (1): 1-36, 1991.

Andreas Weingartner, Practical numbers and the distribution of divisors, Q. J. Math. 66 (2015), 743 - 758.

Andreas Weingartner, On the constant factor in several related asymptotic estimates, arXiv preprint arXiv:1705.06349 [math.NT], 2017-2018.

Andreas Weingartner, The constant factor in the asymptotic for practical numbers, arXiv:1906.07819 [math.NT], 2019.

Formula

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

Mathematica

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 *)

Prog

(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]

Crossrefs

Cf. A005153, A209237, A273773.

Sequence in context: A196471 A265279 A034164 * A103433 A081015 A090139

Adjacent sequences: A322254 A322255 A322256 * A322258 A322259 A322260

Keyword

nonn,more

Author

Amiram Eldar, Dec 01 2018

Status

approved