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

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, 24, 27, 19, 30, 23, 31, 29, 30, 30, 36, 25, 35, 34, 39, 30, 42, 31, 41, 41, 41, 34, 48, 38, 48, 44, 51, 36, 53, 46, 55, 48, 51, 42, 60, 43, 57, 59, 63, 52

Offset: 1

Author

Robert Israel, Jun 04 2015

Keywords

• nonn
• look

Examples

			A000203(7) = 8 >= A000203(k) for k = 1,2,3,4,5,7, so a(7) = 6.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A002093, A007609.

Programs

Maple

N:= 1000:
B:= map(numtheory:-sigma,[$1..N]):
M:= max(B):
X:= Vector(M):
for n from 1 to N do
  b:= B[n];
  X[b..-1]:= X[b..-1] + <(1$(M-b+1))>;
  A[n]:= X[b];
od:
seq(A[n],n=1..N);

Mathematica

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 *)

Formula

a(n) <= n, with equality if and only if n is in A002093.

Empirically it appears that lim inf_(n -> infinity) a(n)/n = 2/3, with minimum value a(29)/29 = 19/29.