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A253248 Number of k <= n with A000203(k) <= A000203(n).

%I A253248 #19 Dec 20 2024 02:43:24
%S A253248 1,2,3,4,4,6,6,8,8,10,8,12,10,13,14,16,13,18,14,20,19,20,17,24,20,25,
%T A253248 24,27,19,30,23,31,29,30,30,36,25,35,34,39,30,42,31,41,41,41,34,48,38,
%U A253248 48,44,51,36,53,46,55,48,51,42,60,43,57,59,63,52
%N A253248 Number of k <= n with A000203(k) <= A000203(n).
%H A253248 Robert Israel, <a href="/A253248/b253248.txt">Table of n, a(n) for n = 1..10000</a>
%F A253248 a(n) <= n, with equality if and only if n is in A002093.
%F A253248 Empirically it appears that lim inf_(n -> infinity) a(n)/n = 2/3, with minimum value a(29)/29 = 19/29.
%e A253248 A000203(7) = 8 >= A000203(k) for k = 1,2,3,4,5,7, so a(7) = 6.
%p A253248 N:= 1000:
%p A253248 B:= map(numtheory:-sigma,[$1..N]):
%p A253248 M:= max(B):
%p A253248 X:= Vector(M):
%p A253248 for n from 1 to N do
%p A253248   b:= B[n];
%p A253248   X[b..-1]:= X[b..-1] + <(1$(M-b+1))>;
%p A253248   A[n]:= X[b];
%p A253248 od:
%p A253248 seq(A[n],n=1..N);
%t A253248 f[v_] := Count[v, _?(# <= v[[-1]] &)]; seq[lim_] := Module[{v = DivisorSigma[1, Range[lim]]}, f[v[[1 ;; #]]] & /@ Range[Length[v]]]; seq[65] (* _Amiram Eldar_, Dec 19 2024 *)
%Y A253248 Cf. A000203, A002093, A007609.
%K A253248 nonn,look
%O A253248 1,2
%A A253248 _Robert Israel_, Jun 04 2015