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

%I A061071 #16 Sep 08 2023 17:31:38
%S A061071 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,
%T A061071 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,
%U A061071 11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11
%N A061071 Number of distinct values in the list of number of divisors, d(j), j=1...n.
%D A061071 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.
%H A061071 T. D. Noe, <a href="/A061071/b061071.txt">Table of n, a(n) for n = 1..10000</a>
%H A061071 P. Erdős and L. Mirsky, <a href="http://www.renyi.hu/~p_erdos/1952-12.pdf">The distribution of values of the divisor function d(n)</a>, Proc. London Math. Soc. 2 (1952), pp. 257-271.
%F A061071 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
%t A061071 a[n_] = Length[Union[Table[DivisorSigma[0, w], {w, 1, n}]]]
%o A061071 (PARI) v=[];vector(100,n,t=numdiv(n);v=vecsort(concat(v,t),,8);#v) \\ _Charles R Greathouse IV_, Dec 12 2012
%o A061071 (Python)
%o A061071 from sympy import divisor_count
%o A061071 def A061071(n): return len({divisor_count(i) for i in range(1,n+1)}) # _Chai Wah Wu_, Sep 08 2023
%Y A061071 Cf. A000005.
%K A061071 nonn
%O A061071 1,2
%A A061071 _Labos Elemer_, May 28 2001