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%I #28 Apr 17 2024 02:31:15
%S 1,3,5,7,9,11,13,15,17,19,21,24,26,28,30,32,34,37,39,42,44,46,48,51,
%T 53,55,57,60,62,65,67,70,72,74,76,78,80,82,84,87,89,92,94,97,100,102,
%U 104,107,109,112,114,117,119,122,124,127,129,131,133,137,139,141,144,146,148,151,153,156,158
%N a(n) = Sum_{k = 1..n} d(d(d(k))), where d(k) is the number of divisors of k (A000005).
%H David A. Corneth, <a href="/A293167/b293167.txt">Table of n, a(n) for n = 1..10000</a>
%H Richard Bellman and Harold N. Shapiro, <a href="http://www.jstor.org/stable/1969281">On a problem in additive number theory</a>, Annals Math., 49 (1948), 333-340.
%H Imre Kátai, <a href="https://doi.org/10.5486/PMD.1969.16.1-4.02">On the iteration of the divisor-function</a>, Publ. Math. Debrecen, Vol. 16 (1969), pp. 3-15.
%F a(1) = 1; a(n + 1) = a(n) + A036450(n + 1) for n > 0. - _David A. Corneth_, Oct 17 2017
%F a(n) = (1 + o(1)) * c * n * log(log(log(n))), where c > 0 is a constant (Kátai, 1969). - _Amiram Eldar_, Apr 17 2024
%t Accumulate[Table[DivisorSigma[0, DivisorSigma[0, DivisorSigma[0, n]]], {n, 80}]] (* _Alonso del Arte_, Oct 17 2017 *)
%o (PARI) a(n) = sum(k=1, n, numdiv(numdiv(numdiv(k)))); \\ _Michel Marcus_, Oct 17 2017
%o (PARI) first(n) = {my(v = vector(n)); v[1] = 1; for(i=2,n,v[i] = v[i-1] + numdiv(numdiv(numdiv(i)))); v} \\ _David A. Corneth_, Oct 17 2017
%Y Part of the sequence A000005, A006218, A010553, A036450, A139130.
%K nonn,changed
%O 1,2
%A _N. J. A. Sloane_, Oct 17 2017
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