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A139130 a(n) = Sum_{k=1..n} d(d(k)), where d(k) = number of divisors of k.

%I A139130 #21 Jan 15 2024 01:46:04
%S A139130 1,3,5,7,9,12,14,17,19,22,24,28,30,33,36,38,40,44,46,50,53,56,58,62,
%T A139130 64,67,70,74,76,80,82,86,89,92,95,98,100,103,106,110,112,116,118,122,
%U A139130 126,129,131,135,137,141,144,148,150,154,157,161,164,167,169,175,177,180
%N A139130 a(n) = Sum_{k=1..n} d(d(k)), where d(k) = number of divisors of k.
%H A139130 Alois P. Heinz, <a href="/A139130/b139130.txt">Table of n, a(n) for n = 1..1000</a>
%H A139130 Richard Bellman and Harold N. Shapiro, <a href="https://www.jstor.org/stable/1969281">On a problem in additive number theory</a>, Annals Math., Vol. 49, No. 2 (1948), 333-340. See Eq. 1.6. [From _N. J. A. Sloane_, Mar 12 2009]
%H A139130 Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1969-12.pdf">On the sum Sum_{n=1..x} d[d(n)]</a>, Math. Student, Vol. 36 (1968), pp. 227-229.
%H A139130 E. Heppner, <a href="https://eudml.org/doc/151410">Über die Iteration von Teilerfunktionen</a>, Journal für die reine und angewandte Mathematik, Vol. 265 (1974), pp. 176-182.
%F A139130 a(n) = b * n * log(log(n)) +  Sum_{k=0..floor(sqrt(n))} b_k * n/log(n)^k  + O(n * exp(-c*sqrt(log(n)))), where b, b_k and c are constants (Heppner, 1974). - _Amiram Eldar_, Jan 15 2024
%p A139130 with(numtheory): a:= n-> add(tau(tau (k)), k=1..n): seq(a(n), n=1..70); # _Alois P. Heinz_, Aug 28 2008
%t A139130 Table[Sum[DivisorSigma[0,DivisorSigma[0,k]],{k,1,n}],{n,1,62}] (* _Geoffrey Critzer_, Sep 28 2013 *)
%t A139130 Accumulate[Table[DivisorSigma[0, DivisorSigma[0, k]], {k, 1, 62}]] (* _Amiram Eldar_, Jan 15 2024 *)
%o A139130 (PARI) a(n) = sum(k = 1, n, numdiv(numdiv(k))); \\ _Michel Marcus_, Sep 28 2013
%Y A139130 Cf. A000005, A010553, A006218.
%K A139130 nonn,easy
%O A139130 1,2
%A A139130 _Leroy Quet_, Jun 05 2008
%E A139130 More terms from _Alois P. Heinz_, Aug 28 2008