A010553 - OEIS (original) (raw)

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, 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, 3, 4, 2, 6, 2, 3, 4, 4, 3, 4, 2, 4, 2

COMMENTS

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).

Every number eventually appears. Sequence A193987 gives the least term where each number appears. - T. D. Noe, Aug 10 2011

REFERENCES

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.

FORMULA

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

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.].

MAPLE

with(numtheory): f := n->tau(tau(n));

MATHEMATICA

Table[Nest[DivisorSigma[0, #] &, n, 2], {n, 81}] (* Michael De Vlieger, Dec 24 2015 *)