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%I #12 Feb 17 2024 11:27:52
%S 1,2,3,4,6,7,8,11,12,14,15,16,20,22,23,24,27,30,31,32,40,43,44,47,48,
%T 52,54,59,60,62,63,70,71,72,78,79,80,86,87,88,92,94,95,96,104,107,108,
%U 116,119,120,123,124,128,135,139,140,142,143,144,152,155,156,158
%N Numbers m such that k + A005361(k) <= m for all k < m.
%C Erdős (1979) proved that the asymptotic density of this sequence is positive.
%C The numbers of terms not exceeding 10^k for k = 1, 2, ... are 7, 44, 307, 2778, 26808, 265339, 2645683, 26433775, 264269957, 2642484069, ... Apparently the asymptotic density of this sequence is about 0.2642...
%D József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004. See chapter 4, p. 333.
%H Amiram Eldar, <a href="/A336444/b336444.txt">Table of n, a(n) for n = 1..10000</a>
%H Paul Erdős, <a href="https://doi.org/10.1007/BF01903382">Some unconventional problems in number theory</a>, Acta Mathematica Academiae Scientiarum Hungarica, Vol. 33, No. 1-2 (1979), pp. 71-80, <a href="https://users.renyi.hu/~p_erdos/1979-23.pdf">alternative link</a>.
%e 3 is a term since 1 + A005361(1) = 2 and 2 + A005361(2) = 3 do not exceed 3.
%t b[1] = 1; b[n_] := Times @@ FactorInteger[n][[;; , 2]]; f[n_] := n + b[n]; fm = 0; s = {1}; Do[fm = Max[fm, f[n]]; If[n + 1 >= fm, AppendTo[s, n + 1]], {n, 1, 160}]; s
%Y Cf. A005361.
%K nonn
%O 1,2
%A _Amiram Eldar_, Jul 21 2020
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