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 A336444 Numbers m such that k + A005361(k) <= m for all k < m. 1

%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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Last modified August 14 04:13 EDT 2024. Contains 375146 sequences. (Running on oeis4.)