A061071 Number of distinct values in the list of number of divisors, d(j), j=1...n.

1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Author

Labos Elemer, May 28 2001

Keywords

• nonn

References

• B. Spearman and K. S. Williams, Handbook of Estimates in the Theory of Numbers, Carleton Math. Lecture Note Series No. 14, 1975; see p. 2.2.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• P. Erdős and L. Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc. 2 (1952), pp. 257-271.

Crossrefs

Cf. A000005.

Programs

Mathematica

a[n_] = Length[Union[Table[DivisorSigma[0, w], {w, 1, n}]]]

PARI

v=[];vector(100,n,t=numdiv(n);v=vecsort(concat(v,t),,8);#v) \\ Charles R Greathouse IV, Dec 12 2012

Python

from sympy import divisor_count
def A061071(n): return len({divisor_count(i) for i in range(1,n+1)}) # Chai Wah Wu, Sep 08 2023

Formula

Erdős & Mirsky show that log a(n) ~ k sqrt(log x)/log log x where k = Pi sqrt(8/3). - Charles R Greathouse IV, Dec 07 2012