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 A070319 Max( tau(k) : k=1,2,3,...,n ) where tau(n)=A000005(n) is the number of divisors of x. 8
 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 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, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Is this the same as A068509? - David Scambler, Sep 10 2012 They are different even asymptotically: A068509(n)=O(sqrt(n)), while a(n) does not have polynomial growth. One example where the sequences differ: a(625) = 24 < A068509(625). (The inequality is implied by the set {1,2,..,25} where each pair of the elements has lcm <= 625.) - Max Alekseyev, Sep 11 2012 The two sequences first differ when n = 336, due to the set of 21 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 21, 24, 30, 36, 42, 48} where each pair of elements has lcm <= 336, while no positive integer <= 336 has more than 20 divisors. Therefore A068509(336) = 21 and A070319(336) = 20. - William Rex Marshall, Sep 11 2012 REFERENCES Sándor, J., Crstici, B., Mitrinović, Dragoslav S. Handbook of Number Theory I. Dordrecht: Kluwer Academic, 2006, p. 44. S. Wigert. Sur l’ordre de grandeur du nombre des diviseurs d’un entier. Arkiv. for Math. 3 (1907), 1-9. LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409. FORMULA a(n) = exp(log(2) log(n) / log(log(n)) + O(log(n) log(log(log(n))) / (log(log(n)))^2)). (See Sándor reference for more formulas.) - Eric M. Schmidt, Jun 30 2013 a(n) = A002183(A261100(n)). - Antti Karttunen, Jun 06 2017 MATHEMATICA a = {0}; Do[AppendTo[a, Max[DivisorSigma[0, n], a[[n]]]], {n, 120}]; Rest@ a (* Michael De Vlieger, Sep 29 2015 *) PROG (PARI) a(n)=vecmax(vector(n, k, numdiv(k))) (PARI) v=vector(100); v=1; for(n=2, #v, v[n]=max(v[n-1], numdiv(n))); v \\ Charles R Greathouse IV, Sep 12 2012 (Haskell) a070319 n = a070319_list !! (n-1) a070319_list = scanl1 max \$ map a000005 [1..] -- Reinhard Zumkeller, Apr 01 2011 (PARI) A070319(n, m=1, s=2)={for(k=s, n, m

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Last modified July 22 07:23 EDT 2019. Contains 325216 sequences. (Running on oeis4.)