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

1,2

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.

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000005.

Sequence in context: A372556 A130249 A286105 * A122258 A332220 A263089

Adjacent sequences: A061068 A061069 A061070 * A061072 A061073 A061074

Keyword

nonn

Author

Labos Elemer, May 28 2001

Status

approved