A138009 a(n) = number of positive integers k, k <= n, where d(k) >= d(n); d(n) = number of positive divisors of n.

1, 1, 2, 1, 4, 1, 6, 2, 4, 3, 10, 1, 12, 5, 6, 2, 16, 2, 18, 3, 10, 11, 22, 1, 15, 13, 14, 5, 28, 2, 30, 7, 18, 19, 20, 1, 36, 22, 23, 4, 40, 5, 42, 11, 12, 28, 46, 1, 33, 14, 31, 15, 52, 7, 34, 8, 36, 37, 58, 1, 60, 39, 19, 10, 42, 10, 66, 22, 45, 11, 70, 2, 72, 48, 25, 26, 51, 13, 78, 4

Offset: 1

Author

Leroy Quet, Feb 27 2008

Keywords

• nonn
• look

Examples

			9 has 3 positive divisors. Among the first 9 positive integers, there are four that have more than or equal the number of divisors than 9 has: 4, with 3 divisors; 6, with 4 divisors; 8, with 4 divisors; and 9, with 3 divisors. So a(9) = 4.
		

Links

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

Crossrefs

Cf. A000005, A002182, A067004, A079788.

Programs

Maple

L:= [2]: A[1]:= 1:
for n from 2 to 100 do
  v:= 2*numtheory:-tau(n);
  k:= ListTools:-BinaryPlace(L,v-1);
  A[n]:= n-k;
  L:= [op(L[1..k]),v,op(L[k+1..-1])];
od:
seq(A[i],i=1..100); # Robert Israel, Sep 26 2018

Mathematica

Table[Length[Select[Range[n], Length[Divisors[ # ]]>=Length[Divisors[n]]&]], {n,1,100}] (* Stefan Steinerberger, Feb 29 2008 *)

PARI

a(n) = my(dn=numdiv(n)); sum(k=1, n, numdiv(k) >= dn); \\ Michel Marcus, Sep 26 2018

Formula

From Amiram Eldar, Jun 26 2025: (Start)

a(n) = n - 1 if and only if n is prime.

a(n) = 1 if and only if n is a highly composite number (A002182). (End)

Extensions

More terms from Stefan Steinerberger, Feb 29 2008