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A061071
Number of distinct values in the list of number of divisors, d(j), j=1...n.
1
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
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
KEYWORD
nonn
AUTHOR
Labos Elemer, May 28 2001
STATUS
approved