

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
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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. 257271.


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


CROSSREFS

Cf. A000005.
Sequence in context: A261100 A130249 A286105 * A122258 A332220 A263089
Adjacent sequences: A061068 A061069 A061070 * A061072 A061073 A061074


KEYWORD

nonn


AUTHOR

Labos Elemer, May 28 2001


STATUS

approved



