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%I #58 Nov 03 2023 20:24:40
%S 1,2,2,2,2,3,2,3,2,3,2,4,2,3,3,2,2,4,2,4,3,3,2,4,2,3,3,4,2,4,2,4,3,3,
%T 3,3,2,3,3,4,2,4,2,4,4,3,2,4,2,4,3,4,2,4,3,4,3,3,2,6,2,3,4,2,3,4,2,4,
%U 3,4,2,6,2,3,4,4,3,4,2,4,2
%N a(n) = tau(tau(n)).
%C Ramanujan (1915) posed the problem of finding the extreme large values of a(n). Buttkewitz et al. determined the maximal order of log a(n).
%C Every number eventually appears. Sequence A193987 gives the least term where each number appears. - _T. D. Noe_, Aug 10 2011
%D S. Ramanujan, Highly composite numbers. Proc. London Math. Soc., series 2, 14 (1915), 347-409. Republished in Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI, 2000, pp. 78-128.
%H Indranil Ghosh, <a href="/A010553/b010553.txt">Table of n, a(n) for n = 1..10000</a> (first 2000 terms from Enrique Pérez Herrero)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
%H R. Bellman and H. N. Shapiro, <a href="http://www.jstor.org/stable/1969281">On a problem in additive number theory</a>, Annals Math., 49 (1948), 333-340. See Eq. 1.5.
%H Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford, Jan-Christoph Schlage-Puchta, <a href="http://arxiv.org/abs/1108.1815">A problem of Ramanujan, Erdos and Katai on the iterated divisor function</a>, arXiv:1108.1815 [math.NT], Aug 08 2011.
%F a(n) = A000005(A000005(n)). a(1) = 1, a(p) = 2 for p = primes (A000040), a(pq) = 3 for pq = product of two distinct primes (A006881), a(pq...z) = k + 1 for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944), a(p^k) = A000005(k+1) for p^k = prime powers (A000961(n) for n > 1), k = natural numbers (A000027). - _Jaroslav Krizek_, Jul 17 2009
%F a(A007947(n)) = 1 + A001221(n); (n>1). - _Enrique Pérez Herrero_, May 30 2010
%F Asymptotically, Max_{i<=n} log(tau(tau(i))) = sqrt(log(n))/log_2(n) * (c + O(log_3(n)/log_2(n)) where c = 8*Sum_{j>=1} log^2 (1 + 1/j)) ~ 2.7959802335... [Buttkewitz et al.].
%p with(numtheory): f := n->tau(tau(n));
%t Table[Nest[DivisorSigma[0, #] &, n, 2], {n, 81}] (* _Michael De Vlieger_, Dec 24 2015 *)
%o (PARI) A010553(n)=numdiv(numdiv(n)); \\ _Enrique Pérez Herrero_, Jul 13 2010
%Y Cf. A000005, A036450, A193987 (least number k such that tau(tau(k)) = n), A335831.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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